A292866 -id:A292866 - cp's OEIS frontend

A242817 a(n) = B(n,n), where B(n,x) = Sum_{k=0..n} Stirling2(n,k)*x^k are the Bell polynomials (also known as exponential polynomials or Touchard polynomials).

1, 1, 6, 57, 756, 12880, 268098, 6593839, 187104200, 6016681467, 216229931110, 8588688990640, 373625770888956, 17666550789597073, 902162954264563306, 49482106424507339565, 2901159958960121863952, 181069240855214001514460, 11985869691525854175222222

Offset: 0

Emanuele Munarini, May 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..370
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Cf. A035051, A292866, A346654, A346655.

Main diagonal of A189233 and of A292860.

A:= proc(n, k) option remember; `if`(n=0, 1, (1+
      add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
a:= n-> A(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, May 17 2016

Mathematica
```
Table[BellB[n, n], {n, 0, 100}]
```

a(n):=stirling2(n,0)+sum(stirling2(n,k)*n^k,k,1,n);
makelist(a(n),n,0,30);

a(n) = sum(k=0, n, stirling(n,k,2)*n^k); \\ Michel Marcus, Apr 20 2016

E.g.f.: x*f'(x)/f(x), where f(x) is the generating series for sequence A035051.
a(n) ~ (exp(1/LambertW(1)-2)/LambertW(1))^n * n^n / sqrt(1+LambertW(1)). - Vaclav Kotesovec, May 23 2014
Conjecture: It appears that the equation a(x)*e^x = Sum_{n=0..oo} ( (n^x*x^n)/n! ) is true for every positive integer x. - Nicolas Nagel, Apr 20 2016 [This is just the special case k=x of the formula  B(k,x) = e^(-x) * Sum_{n=0..oo} n^k*x^n/n!; see for example the World of Mathematics link. - Pontus von Brömssen, Dec 05 2020]
a(n) = n! * [x^n] exp(n*(exp(x)-1)). - Alois P. Heinz, May 17 2016
a(n) = [x^n] Sum_{k=0..n} n^k*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, May 31 2018

Name corrected by Pontus von Brömssen, Dec 05 2020

A292861 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*(1 - exp(x))).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, 1, 0, 1, -4, 6, 2, 1, 0, 1, -5, 12, -3, -6, -2, 0, 1, -6, 20, -20, -21, -14, -9, 0, 1, -7, 30, -55, -20, 24, 26, -9, 0, 1, -8, 42, -114, 45, 172, 195, 178, 50, 0, 1, -9, 56, -203, 246, 370, 108, -111, 90, 267, 0, 1, -10, 72, -328, 679, 318, -1105, -2388, -3072, -2382, 413, 0

Offset: 0

Seiichi Manyama, Sep 25 2017

			Square array begins:
   1,  1,   1,   1,   1,     1,     1, ...
   0, -1,  -2,  -3,  -4,    -5,    -6, ...
   0,  0,   2,   6,  12,    20,    30, ...
   0,  1,   2,  -3, -20,   -55,  -114, ...
   0,  1,  -6, -21, -20,    45,   246, ...
   0, -2, -14,  24, 172,   370,   318, ...
   0, -9,  26, 195, 108, -1105, -4074, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A000007, A000587, A213170, A309084, A309085.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292866.
Cf. A292860, A309386.

A:= proc(n, k) option remember; `if`(n=0, 1,
      -(1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

A[n_, k_] := Sum[(-k)^j StirlingS2[n, j], {j, 0, n}];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 10 2021 *)
A292861[n_, k_] := BellB[k, k - n];
Table[A292861[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)

A(0,k) = 1 and A(n,k) = -k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} (-k)^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
A(n,k) = BellPolynomial(n, -k). - Peter Luschny, Dec 23 2021

A350263 Triangle read by rows. T(n, k) = BellPolynomial(n, -k).

Original entry on oeis.org

1, 0, -1, 0, 0, 2, 0, 1, 2, -3, 0, 1, -6, -21, -20, 0, -2, -14, 24, 172, 370, 0, -9, 26, 195, 108, -1105, -4074, 0, -9, 178, -111, -2388, -4805, 2046, 34293, 0, 50, 90, -3072, -3220, 23670, 87510, 111860, -138312, 0, 267, -2382, -4053, 47532, 121995, -115458, -1193157, -2966088, -2932533

Offset: 0

Peter Luschny, Dec 23 2021

			[0] 1
[1] 0,  -1
[2] 0,   0,     2
[3] 0,   1,     2,    -3
[4] 0,   1,    -6,   -21,   -20
[5] 0,  -2,   -14,    24,   172,    370
[6] 0,  -9,    26,   195,   108,  -1105, -  4074
[7] 0,  -9,   178,  -111, -2388,  -4805,    2046,    34293
[8] 0,  50,    90, -3072, -3220,  23670,   87510,   111860,  -138312
[9] 0, 267, -2382, -4053, 47532, 121995, -115458, -1193157, -2966088, -2932533

Main diagonal: A292866, column 1: A000587, variant: A292861.

A350263 := (n, k) -> ifelse(n = 0, 1, BellB(n, -k)):
seq(seq(A350263(n, k), k = 0..n), n = 0..9);

T[n_, k_] := BellB[n, -k]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + njx).

Original entry on oeis.org

1, 1, -1, 1, 25, -674, 15211, -331827, 5987745, 15901597, -13125035449, 1292056076070, -103145930581319, 7462324963409941, -464957409070517453, 16313974895147212801, 2059903411953959582849, -708700955022151333496910, 143215213612865558214820303, -24681846509158429152517973103

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..282
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A009235, A014182, A292866, A301419, A317996, A318179, A318180, A318181.

Table[SeriesCoefficient[Sum[x^k/Product[(1 + n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(1 - Exp[-n x])/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[(-n)^(n - k) StirlingS2[n, k], {k, n}], {n, 19}]]
Join[{1}, Table[(-n)^n BellB[n, -1/n], {n, 1, 21}]] (* Peter Luschny, Aug 20 2018 *)

{a(n) = sum(k=0, n, (-n)^(n-k)*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 27 2019

a(n) = n! * [x^n] exp((1 - exp(-n*x))/n), for n > 0.
a(n) = Sum_{k=0..n} (-n)^(n-k)*Stirling2(n,k).
a(n) = (-n)^n*BellPolynomial_n(-1/n) for n >= 1. - Peter Luschny, Aug 20 2018

A334241 a(n) = exp(n) * Sum_{k>=0} (k + 1)^n * (-n)^k / k!.

Original entry on oeis.org

1, 0, -1, 7, -43, 221, -341, -15980, 370761, -5688125, 62689871, -197586839, -14973562979, 585250669316, -14306382821485, 240985102271971, -1121421968408303, -122020498882279931, 6674724196051810807, -223424819176020519168, 5051515662105879438501

Offset: 0

Ilya Gutkovskiy, Apr 19 2020

sign

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A292866, A334193, A334240, A334242, A334243.

Table[n! SeriesCoefficient[Exp[x + n (1 - Exp[x])], {x, 0, n}], {n, 0, 20}]
Table[Sum[Binomial[n, k] BellB[k, -n], {k, 0, n}], {n, 0, 20}]

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (-n*x/(1 - x))^k / Product_{j=1..k} (1 - j*x/(1 - x)).
a(n) = n! * [x^n] exp(x + n*(1 - exp(x))).
a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(-n).

A335868 a(n) = exp(n) * Sum_{k>=0} (-n)^k * (k - 1)^n / k!.

Original entry on oeis.org

1, -2, 7, -31, 149, -631, 475, 43210, -844727, 10960505, -86569889, -584746911, 46302579229, -1304510879686, 25366896568707, -277053418780891, -4271166460501743, 384590020131637825, -14617527176248527545, 380117694164438489422, -5265650620303861935579

Offset: 0

Ilya Gutkovskiy, Jun 27 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A109747, A292866, A334241, A335867.

Table[n! SeriesCoefficient[Exp[n (1 - Exp[x]) - x], {x, 0, n}], {n, 0, 20}]
Table[Sum[(-1)^(n - k) Binomial[n, k] BellB[k, -n], {k, 0, n}], {n, 0, 20}]

a(n) = n! * [x^n] exp(n*(1 - exp(x)) - x).

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * BellPolynomial_k(-n).

A334243 a(n) = exp(n) * Sum_{k>=0} (k + n)^n * (-n)^k / k!.

Original entry on oeis.org

1, 0, -2, -3, 44, 245, -2346, -33278, 186808, 6888555, -6774910, -1986368439, -10227075420, 738830661296, 10363304656782, -327255834908715, -9380517430358288, 152180429032236325, 9132761207739810618, -46897839494116200918, -9833058047657527541220

Offset: 0

Ilya Gutkovskiy, Apr 19 2020

sign

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A292866, A298373, A334240, A334241, A334242.

Table[n! SeriesCoefficient[Exp[n (1 + x - Exp[x])], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[k, -n] n^(n - k), {k, 0, n}], {n, 1, 20}]]

a(n) = n! * [x^n] exp(n*(1 + x - exp(x))).

a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(-n) * n^(n-k).

A334986 a(n) = exp(n) * Sum_{k>=0} (-1)^k * n^(k-1) * k^(n-1) / k!.

Original entry on oeis.org

1, -1, 2, -5, 9, 53, -1107, 12983, -116470, 560049, 8370713, -346902877, 7551856337, -117404648467, 913399734614, 22560135521007, -1393700803877939, 44331044030953865, -979905458659247779, 10462396536804802459, 367799071887303276422, -30046998012662824941947

Offset: 1

Ilya Gutkovskiy, May 18 2020

sign

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A030019, A052888, A292866.

Table[Sum[(-1)^k StirlingS2[n - 1, k] n^(k - 1), {k, 0, n - 1}], {n, 1, 22}]
Table[BellB[n - 1, -n]/n, {n, 1, 22}]

a(n)={sum(k=0, n-1, (-1)^k * stirling(n-1,k,2) * n^(k-1))} \\ Andrew Howroyd, May 18 2020

a(n) = Sum_{k=0..n-1} (-1)^k * Stirling2(n-1,k) * n^(k-1).

a(n) = BellPolynomial_(n-1)(-n) / n.

A334258 a(n) = (-1)^n * exp(n) * Sum_{k>=1} (-1)^k * n^(k-1) * k^n / k!.

Original entry on oeis.org

1, 1, 1, -5, -74, -679, -4899, -17289, 325837, 10627109, 199348590, 2684041427, 15872610469, -546948563407, -27499774835519, -778467357484561, -15311413773551790, -125363405319188419, 6452292137017871097, 436442148982835915339, 16494863323310244977581

Offset: 1

Ilya Gutkovskiy, Apr 20 2020

sign

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A002741, A052888, A292866.

nmax = 21; CoefficientList[InverseSeries[Series[-Log[1 - x] Exp[-x], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest
Table[Sum[(-1)^(n - k) StirlingS2[n, k] n^(k - 1), {k, 1, n}], {n, 1, 21}]
Table[(-1)^n BellB[n, -n]/n, {n, 1, 21}]

a(n) = sum(k=1, n, (-1)^(n-k) * stirling(n,k,2) * n^(k-1)); \\ Michel Marcus, Apr 20 2020

E.g.f.: series reversion of -log(1 - x) * exp(-x).
a(n) = (n - 1)! * [x^n] exp(n*(1 - exp(-x))).
a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling2(n,k) * n^(k-1).
a(n) = (-1)^n * BellPolynomial_n(-n) / n.

cp's OEIS Frontend

A242817 a(n) = B(n,n), where B(n,x) = Sum_{k=0..n} Stirling2(n,k)*x^k are the Bell polynomials (also known as exponential polynomials or Touchard polynomials).

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A292861 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*(1 - exp(x))).

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A350263 Triangle read by rows. T(n, k) = BellPolynomial(n, -k).

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A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + n*j*x).

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A334241 a(n) = exp(n) * Sum_{k>=0} (k + 1)^n * (-n)^k / k!.

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A335868 a(n) = exp(n) * Sum_{k>=0} (-n)^k * (k - 1)^n / k!.

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A334243 a(n) = exp(n) * Sum_{k>=0} (k + n)^n * (-n)^k / k!.

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A334986 a(n) = exp(n) * Sum_{k>=0} (-1)^k * n^(k-1) * k^(n-1) / k!.

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A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + njx).