Stirling numbers

Symbol: StirlingCycle —

\left[{n \atop k}\right]

— Unsigned Stirling number of the first kind

Domain	Codomain
$n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}$	$\left[{n \atop k}\right] \in \mathbb{Z}_{\ge 0}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("778fa2"),
    SymbolDefinition(StirlingCycle, StirlingCycle(n, k), "Unsigned Stirling number of the first kind"),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(0))), Element(StirlingCycle(n, k), ZZGreaterEqual(0))))))

2e9d0c

Symbol: StirlingS1 —

s\!\left(n, k\right)

— Signed Stirling number of the first kind

Domain	Codomain
$n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}$	$s\!\left(n, k\right) \in \mathbb{Z}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`StirlingS1`	$s\!\left(n, k\right)$	Signed Stirling number of the first kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("2e9d0c"),
    SymbolDefinition(StirlingS1, StirlingS1(n, k), "Signed Stirling number of the first kind"),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(0))), Element(StirlingS1(n, k), ZZ)))))

4c6c43

Symbol: StirlingS2 —

\left\{{n \atop k}\right\}

— Stirling number of the second kind

Domain	Codomain
$n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}$	$\left\{{n \atop k}\right\} \in \mathbb{Z}_{\ge 0}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`StirlingS2`	$\left\{{n \atop k}\right\}$	Stirling number of the second kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("4c6c43"),
    SymbolDefinition(StirlingS2, StirlingS2(n, k), "Stirling number of the second kind"),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(0))), Element(StirlingS2(n, k), ZZGreaterEqual(0))))))

1706bb

Symbol: BellNumber —

B_{n}

— Bell number

Domain	Codomain
$n \in \mathbb{Z}_{\ge 0}$	$B_{n} \in \mathbb{Z}_{\ge 1}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`BellNumber`	$B_{n}$	Bell number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("1706bb"),
    SymbolDefinition(BellNumber, BellNumber(n), "Bell number"),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(Element(n, ZZGreaterEqual(0)), Element(BellNumber(n), ZZGreaterEqual(1))))))

f88455

Table of

\left[{n \atop k}\right]

for

0 \le n \le 10

and

0 \le k \le 10

$n$ \ $k$	0	1	2	3	4	5	6	7	8	9	10
0	1	0	0	0	0	0	0	0	0	0	0
1	0	1	0	0	0	0	0	0	0	0	0
2	0	1	1	0	0	0	0	0	0	0	0
3	0	2	3	1	0	0	0	0	0	0	0
4	0	6	11	6	1	0	0	0	0	0	0
5	0	24	50	35	10	1	0	0	0	0	0
6	0	120	274	225	85	15	1	0	0	0	0
7	0	720	1764	1624	735	175	21	1	0	0	0
8	0	5040	13068	13132	6769	1960	322	28	1	0	0
9	0	40320	109584	118124	67284	22449	4536	546	36	1	0
10	0	362880	1026576	1172700	723680	269325	63273	9450	870	45	1

Table data:

\left(n, k, y\right)

such that

\left[{n \atop k}\right] = y

Definitions:

Fungrim symbol	Notation	Short description
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind

Source code for this entry:

Entry(ID("f88455"),
    Description("Table of", StirlingCycle(n, k), "for", LessEqual(0, n, 10), "and", LessEqual(0, k, 10)),
    Table(TableRelation(Tuple(n, k, y), Equal(StirlingCycle(n, k), y)), TableHeadings(Description(n, "\", k), 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), TableColumnHeadings(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), List(Tuple(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 2, 3, 1, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 6, 11, 6, 1, 0, 0, 0, 0, 0, 0), Tuple(0, 24, 50, 35, 10, 1, 0, 0, 0, 0, 0), Tuple(0, 120, 274, 225, 85, 15, 1, 0, 0, 0, 0), Tuple(0, 720, 1764, 1624, 735, 175, 21, 1, 0, 0, 0), Tuple(0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 0, 0), Tuple(0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0), Tuple(0, 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1))))

a93679

Table of

s\!\left(n, k\right)

for

0 \le n \le 10

and

0 \le k \le 10

$n$ \ $k$	0	1	2	3	4	5	6	7	8	9	10
0	1	0	0	0	0	0	0	0	0	0	0
1	0	1	0	0	0	0	0	0	0	0	0
2	0	-1	1	0	0	0	0	0	0	0	0
3	0	2	-3	1	0	0	0	0	0	0	0
4	0	-6	11	-6	1	0	0	0	0	0	0
5	0	24	-50	35	-10	1	0	0	0	0	0
6	0	-120	274	-225	85	-15	1	0	0	0	0
7	0	720	-1764	1624	-735	175	-21	1	0	0	0
8	0	-5040	13068	-13132	6769	-1960	322	-28	1	0	0
9	0	40320	-109584	118124	-67284	22449	-4536	546	-36	1	0
10	0	-362880	1026576	-1172700	723680	-269325	63273	-9450	870	-45	1

Table data:

\left(n, k, y\right)

such that

s\!\left(n, k\right) = y

Definitions:

Fungrim symbol	Notation	Short description
`StirlingS1`	$s\!\left(n, k\right)$	Signed Stirling number of the first kind

Source code for this entry:

Entry(ID("a93679"),
    Description("Table of", StirlingS1(n, k), "for", LessEqual(0, n, 10), "and", LessEqual(0, k, 10)),
    Table(TableRelation(Tuple(n, k, y), Equal(StirlingS1(n, k), y)), TableHeadings(Description(n, "\", k), 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), TableColumnHeadings(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), List(Tuple(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0), Tuple(0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 2, -3, 1, 0, 0, 0, 0, 0, 0, 0), Tuple(0, -6, 11, -6, 1, 0, 0, 0, 0, 0, 0), Tuple(0, 24, -50, 35, -10, 1, 0, 0, 0, 0, 0), Tuple(0, -120, 274, -225, 85, -15, 1, 0, 0, 0, 0), Tuple(0, 720, -1764, 1624, -735, 175, -21, 1, 0, 0, 0), Tuple(0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1, 0, 0), Tuple(0, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1, 0), Tuple(0, -362880, 1026576, -1172700, 723680, -269325, 63273, -9450, 870, -45, 1))))

cecede

Table of

\left\{{n \atop k}\right\}

for

0 \le n \le 10

and

0 \le k \le 10

$n$ \ $k$	0	1	2	3	4	5	6	7	8	9	10
0	1	0	0	0	0	0	0	0	0	0	0
1	0	1	0	0	0	0	0	0	0	0	0
2	0	1	1	0	0	0	0	0	0	0	0
3	0	1	3	1	0	0	0	0	0	0	0
4	0	1	7	6	1	0	0	0	0	0	0
5	0	1	15	25	10	1	0	0	0	0	0
6	0	1	31	90	65	15	1	0	0	0	0
7	0	1	63	301	350	140	21	1	0	0	0
8	0	1	127	966	1701	1050	266	28	1	0	0
9	0	1	255	3025	7770	6951	2646	462	36	1	0
10	0	1	511	9330	34105	42525	22827	5880	750	45	1

Table data:

\left(n, k, y\right)

such that

\left\{{n \atop k}\right\} = y

Definitions:

Fungrim symbol	Notation	Short description
`StirlingS2`	$\left\{{n \atop k}\right\}$	Stirling number of the second kind

Source code for this entry:

Entry(ID("cecede"),
    Description("Table of", StirlingS2(n, k), "for", LessEqual(0, n, 10), "and", LessEqual(0, k, 10)),
    Table(TableRelation(Tuple(n, k, y), Equal(StirlingS2(n, k), y)), TableHeadings(Description(n, "\", k), 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), TableColumnHeadings(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), List(Tuple(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0), Tuple(0, 1, 7, 6, 1, 0, 0, 0, 0, 0, 0), Tuple(0, 1, 15, 25, 10, 1, 0, 0, 0, 0, 0), Tuple(0, 1, 31, 90, 65, 15, 1, 0, 0, 0, 0), Tuple(0, 1, 63, 301, 350, 140, 21, 1, 0, 0, 0), Tuple(0, 1, 127, 966, 1701, 1050, 266, 28, 1, 0, 0), Tuple(0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 0), Tuple(0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1))))

4c6267

Table of

B_{n}

for

0 \le n \le 40

$n$	$B_{n}$
0	1
1	1
2	2
3	5
4	15
5	52
6	203
7	877
8	4140
9	21147
10	115975
11	678570
12	4213597
13	27644437
14	190899322
15	1382958545
16	10480142147
17	82864869804
18	682076806159
19	5832742205057

$n$	$B_{n}$
20	51724158235372
21	474869816156751
22	4506715738447323
23	44152005855084346
24	445958869294805289
25	4638590332229999353
26	49631246523618756274
27	545717047936059989389
28	6160539404599934652455
29	71339801938860275191172
30	846749014511809332450147
31	10293358946226376485095653
32	128064670049908713818925644
33	1629595892846007606764728147
34	21195039388640360462388656799
35	281600203019560266563340426570
36	3819714729894818339975525681317
37	52868366208550447901945575624941
38	746289892095625330523099540639146
39	10738823330774692832768857986425209
40	157450588391204931289324344702531067

Table data:

\left(n, y\right)

such that

B_{n} = y

Definitions:

Fungrim symbol	Notation	Short description
`BellNumber`	$B_{n}$	Bell number

Source code for this entry:

Entry(ID("4c6267"),
    Description("Table of", BellNumber(n), "for", LessEqual(0, n, 40)),
    Table(TableRelation(Tuple(n, y), Equal(BellNumber(n), y)), TableHeadings(n, BellNumber(n)), TableSplit(2), List(Tuple(0, 1), Tuple(1, 1), Tuple(2, 2), Tuple(3, 5), Tuple(4, 15), Tuple(5, 52), Tuple(6, 203), Tuple(7, 877), Tuple(8, 4140), Tuple(9, 21147), Tuple(10, 115975), Tuple(11, 678570), Tuple(12, 4213597), Tuple(13, 27644437), Tuple(14, 190899322), Tuple(15, 1382958545), Tuple(16, 10480142147), Tuple(17, 82864869804), Tuple(18, 682076806159), Tuple(19, 5832742205057), Tuple(20, 51724158235372), Tuple(21, 474869816156751), Tuple(22, 4506715738447323), Tuple(23, 44152005855084346), Tuple(24, 445958869294805289), Tuple(25, 4638590332229999353), Tuple(26, 49631246523618756274), Tuple(27, 545717047936059989389), Tuple(28, 6160539404599934652455), Tuple(29, 71339801938860275191172), Tuple(30, 846749014511809332450147), Tuple(31, 10293358946226376485095653), Tuple(32, 128064670049908713818925644), Tuple(33, 1629595892846007606764728147), Tuple(34, 21195039388640360462388656799), Tuple(35, 281600203019560266563340426570), Tuple(36, 3819714729894818339975525681317), Tuple(37, 52868366208550447901945575624941), Tuple(38, 746289892095625330523099540639146), Tuple(39, 10738823330774692832768857986425209), Tuple(40, 157450588391204931289324344702531067))))

f0d72c

\left[{n + 1 \atop k}\right] = n \left[{n \atop k}\right] + \left[{n \atop k - 1}\right]

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 1}

TeX:

\left[{n + 1 \atop k}\right] = n \left[{n \atop k}\right] + \left[{n \atop k - 1}\right]

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("f0d72c"),
    Formula(Equal(StirlingCycle(Add(n, 1), k), Add(Mul(n, StirlingCycle(n, k)), StirlingCycle(n, Sub(k, 1))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(1)))))

18ec99

s\!\left(n + 1, k\right) = s\!\left(n, k - 1\right) - n s\!\left(n, k\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 1}

TeX:

s\!\left(n + 1, k\right) = s\!\left(n, k - 1\right) - n s\!\left(n, k\right)

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`StirlingS1`	$s\!\left(n, k\right)$	Signed Stirling number of the first kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("18ec99"),
    Formula(Equal(StirlingS1(Add(n, 1), k), Sub(StirlingS1(n, Sub(k, 1)), Mul(n, StirlingS1(n, k))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(1)))))

9fbe4f

\left\{{n + 1 \atop k}\right\} = k \left\{{n \atop k}\right\} + \left\{{n \atop k - 1}\right\}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 1}

TeX:

\left\{{n + 1 \atop k}\right\} = k \left\{{n \atop k}\right\} + \left\{{n \atop k - 1}\right\}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`StirlingS2`	$\left\{{n \atop k}\right\}$	Stirling number of the second kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("9fbe4f"),
    Formula(Equal(StirlingS2(Add(n, 1), k), Add(Mul(k, StirlingS2(n, k)), StirlingS2(n, Sub(k, 1))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(1)))))

071a94

s\!\left(n, k\right) = {\left(-1\right)}^{n + k} \left[{n \atop k}\right]

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}

TeX:

s\!\left(n, k\right) = {\left(-1\right)}^{n + k} \left[{n \atop k}\right]

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`StirlingS1`	$s\!\left(n, k\right)$	Signed Stirling number of the first kind
`Pow`	${a}^{b}$	Power
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("071a94"),
    Formula(Equal(StirlingS1(n, k), Mul(Pow(-1, Add(n, k)), StirlingCycle(n, k)))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(0)))))

21241f

\left(x\right)_{n} = \sum_{k=0}^{n} \left[{n \atop k}\right] {x}^{k}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}

TeX:

\left(x\right)_{n} = \sum_{k=0}^{n} \left[{n \atop k}\right] {x}^{k}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("21241f"),
    Formula(Equal(RisingFactorial(x, n), Sum(Mul(StirlingCycle(n, k), Pow(x, k)), Tuple(k, 0, n)))),
    Variables(x, n),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

f46e0e

\left(x - n + 1\right)_{n} = \sum_{k=0}^{n} s\!\left(n, k\right) {x}^{k}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}

TeX:

\left(x - n + 1\right)_{n} = \sum_{k=0}^{n} s\!\left(n, k\right) {x}^{k}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`StirlingS1`	$s\!\left(n, k\right)$	Signed Stirling number of the first kind
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f46e0e"),
    Formula(Equal(RisingFactorial(Add(Sub(x, n), 1), n), Sum(Mul(StirlingS1(n, k), Pow(x, k)), Tuple(k, 0, n)))),
    Variables(x, n),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

b823b0

{x}^{n} = \sum_{k=0}^{n} \left\{{n \atop k}\right\} \left(x - n + 1\right)_{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}

TeX:

{x}^{n} = \sum_{k=0}^{n} \left\{{n \atop k}\right\} \left(x - n + 1\right)_{n}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`StirlingS2`	$\left\{{n \atop k}\right\}$	Stirling number of the second kind
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b823b0"),
    Formula(Equal(Pow(x, n), Sum(Mul(StirlingS2(n, k), RisingFactorial(Add(Sub(x, n), 1), n)), Tuple(k, 0, n)))),
    Variables(x, n),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

b01280

\frac{{\left(\log\!\left(1 + x\right)\right)}^{k}}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}

Assumptions:

k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1

TeX:

\frac{{\left(\log\!\left(1 + x\right)\right)}^{k}}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}

k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Log`	$\log\!\left(z\right)$	Natural logarithm
`Factorial`	$n !$	Factorial
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("b01280"),
    Formula(Equal(Div(Pow(Log(Add(1, x)), k), Factorial(k)), Sum(Mul(Mul(Pow(-1, Sub(n, k)), StirlingCycle(n, k)), Div(Pow(x, n), Factorial(n))), Tuple(n, k, Infinity)))),
    Variables(x, k),
    Assumptions(And(Element(k, ZZGreaterEqual(0)), Element(x, CC), Less(Abs(x), 1))))

a9a610

\frac{{\left({e}^{x} - 1\right)}^{k}}{k !} = \sum_{n=k}^{\infty} \left\{{n \atop k}\right\} \frac{{x}^{n}}{n !}

Assumptions:

k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}

TeX:

\frac{{\left({e}^{x} - 1\right)}^{k}}{k !} = \sum_{n=k}^{\infty} \left\{{n \atop k}\right\} \frac{{x}^{n}}{n !}

k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Factorial`	$n !$	Factorial
`StirlingS2`	$\left\{{n \atop k}\right\}$	Stirling number of the second kind
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a9a610"),
    Formula(Equal(Div(Pow(Sub(Exp(x), 1), k), Factorial(k)), Sum(Mul(StirlingS2(n, k), Div(Pow(x, n), Factorial(n))), Tuple(n, k, Infinity)))),
    Variables(x, k),
    Assumptions(And(Element(k, ZZGreaterEqual(0)), Element(x, CC))))

6189b9

\left\{{n \atop k}\right\} = \frac{1}{k !} \sum_{i=0}^{k} {\left(-1\right)}^{i} {k \choose i} {\left(k - i\right)}^{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}

TeX:

\left\{{n \atop k}\right\} = \frac{1}{k !} \sum_{i=0}^{k} {\left(-1\right)}^{i} {k \choose i} {\left(k - i\right)}^{n}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`StirlingS2`	$\left\{{n \atop k}\right\}$	Stirling number of the second kind
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("6189b9"),
    Formula(Equal(StirlingS2(n, k), Mul(Div(1, Factorial(k)), Sum(Mul(Mul(Pow(-1, i), Binomial(k, i)), Pow(Sub(k, i), n)), Tuple(i, 0, k))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(0)))))

ea9e2f

\sum_{k=0}^{n} \left[{n \atop k}\right] = n !

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\sum_{k=0}^{n} \left[{n \atop k}\right] = n !

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ea9e2f"),
    Formula(Equal(Sum(StirlingCycle(n, k), Tuple(k, 0, n)), Factorial(n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

255576

\sum_{k=0}^{n} \left\{{n \atop k}\right\} = B_{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\sum_{k=0}^{n} \left\{{n \atop k}\right\} = B_{n}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`StirlingS2`	$\left\{{n \atop k}\right\}$	Stirling number of the second kind
`BellNumber`	$B_{n}$	Bell number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("255576"),
    Formula(Equal(Sum(StirlingS2(n, k), Tuple(k, 0, n)), BellNumber(n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

7774a3

\left[{n \atop k}\right] \le \frac{{2}^{n} n !}{k !}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}

TeX:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("7774a3"),
    LessEqual(StirlingCycle(n, k), Div(Mul(Pow(2, n), Factorial(n)), Factorial(k))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(0)))))

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