A292860 -id:A292860 - 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

A357681 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * (exp(x) - 1) ).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 8, 0, 1, 0, 4, 9, 18, 25, 0, 1, 0, 5, 12, 30, 70, 97, 0, 1, 0, 6, 15, 44, 135, 330, 434, 0, 1, 0, 7, 18, 60, 220, 705, 1694, 2095, 0, 1, 0, 8, 21, 78, 325, 1228, 3906, 9202, 10707, 0, 1, 0, 9, 24, 98, 450, 1905, 7196, 22953, 53334, 58194, 0

Offset: 0

Seiichi Manyama, Oct 09 2022

			Square array begins:
  1,  1,  1,   1,   1,   1, ...
  0,  0,  0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5, ...
  0,  3,  6,   9,  12,  15, ...
  0,  8, 18,  30,  44,  60, ...
  0, 25, 70, 135, 220, 325, ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Bell Polynomial.

Columns k=0-4 give: A000007, A024430, A264036, A357615, A065143.
Column k=9 gives A357667.
Main diagonal gives A357682.
Cf. A292860.

T(n, k) = sum(j=0, n\2, k^j*stirling(n, 2*j, 2));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
T(n, k) = round((Bell_poly(n, sqrt(k))+Bell_poly(n, -sqrt(k))))/2;

T(n,k) = Sum_{j=0..floor(n/2)} k^j * Stirling2(n,2*j).

T(n,k) = ( Bell_n(sqrt(k)) + Bell_n(-sqrt(k)) )/2, where Bell_n(x) is n-th Bell polynomial.

A350256 Triangle read by rows. T(n, k) = BellPolynomial(n, k).

Original entry on oeis.org

1, 0, 1, 0, 2, 6, 0, 5, 22, 57, 0, 15, 94, 309, 756, 0, 52, 454, 1866, 5428, 12880, 0, 203, 2430, 12351, 42356, 115155, 268098, 0, 877, 14214, 88563, 355636, 1101705, 2869242, 6593839, 0, 4140, 89918, 681870, 3188340, 11202680, 32510850, 82187658, 187104200

Offset: 0

Peter Luschny, Dec 22 2021

			Triangle begins:
[0] 1
[1] 0,    1
[2] 0,    2,     6
[3] 0,    5,    22,     57
[4] 0,   15,    94,    309,     756
[5] 0,   52,   454,   1866,    5428,    12880
[6] 0,  203,  2430,  12351,   42356,   115155,   268098
[7] 0,  877, 14214,  88563,  355636,  1101705,  2869242,  6593839
[8] 0, 4140, 89918, 681870, 3188340, 11202680, 32510850, 82187658, 187104200

Cf. A242817 (main diagonal), A000110 (column 1), A350264 (row sums), A350263 (Bell(n,-k)), A189233 and A292860 (array).

A350256 := (n, k) -> ifelse(n = 0, 1, BellB(n, k)):
seq(seq(A350256(n, k), k = 0..n), n = 0..8);

T[n_, k_] := BellB[n, k]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

A351761 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 12, 21, 0, 1, 4, 24, 102, 148, 0, 1, 5, 40, 279, 1160, 1305, 0, 1, 6, 60, 588, 4332, 16490, 13806, 0, 1, 7, 84, 1065, 11536, 84075, 281292, 170401, 0, 1, 8, 112, 1746, 25220, 282900, 1958058, 5598110, 2403640, 0

Offset: 0

Seiichi Manyama, Feb 18 2022

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  0,    1,     2,     3,      4,      5, ...
  0,    4,    12,    24,     40,     60, ...
  0,   21,   102,   279,    588,   1065, ...
  0,  148,  1160,  4332,  11536,  25220, ...
  0, 1305, 16490, 84075, 282900, 746525, ...

Columns k=0..3 give A000007, A006153, A351762, A351763.
Main diagonal gives A351765.
Cf. A292860, A351776.

T(n, k) = n!*sum(j=0, n, k^(n-j)*(n-j)^j/j!);

T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

E.g.f. of column k: 1/(1 - k*x*exp(x)).

T(0,k) = 1 and T(n,k) = k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

A299824 a(n) = (1/e^n)Sum_{j >= 1} j^n n^j / (j-1)!.

Original entry on oeis.org

2, 22, 309, 5428, 115155, 2869242, 82187658, 2661876168, 96202473183, 3838516103310, 167606767714397, 7949901069639228, 407048805012563038, 22376916254447538882, 1314573505901491675965, 82188946843192555474704, 5448870914168179374456623, 381819805747937892412056342

Offset: 1

Pedro Caceres, Feb 19 2018

nonn

For m>1, A242817(m) and a(m-1) are also the m-th and (m+1)-st terms of the sequences "Number of ways of placing X labeled balls into X unlabeled (but (m-1)-colored) boxes". For instance, sequence A144180 for 5-colored boxes (m = 6), has A144180(6) = 12880, and A144180(7) = 115155, which are A242817(6) and a(5) respectively. Same pattern can be observed for A027710, A144223, A144263 (comment added after Omar E. Pol's formula).

			a(4) = (1/e^4)*Sum_{j >= 1} j^4 * 4^j / (j-1)! = 5428.

Alois P. Heinz, Table of n, a(n) for n = 1..368

Cf. A027710, A144180, A144223, A144263, A189233, A242817, A292860.

a(n) = round(exp(-n)*suminf(j = 1, (j^n)*(n^j)/(j-1)!)); \\ Michel Marcus, Feb 24 2018

A299824(n,f=exp(n),S=n/f,t)=for(j=2,oo,S+=(t=j^n*n^j)/(f*=j-1);tn&&return(ceil(S))) \\ For n > 23, use \p## with some ## >= 2n. - M. F. Hasler, Mar 09 2018

a(n) = A189233(n+1,n). - Omar E. Pol, Feb 24 2018
a(n) ~ exp(n/LambertW(1) - 2*n) * n^(n + 1) / (sqrt(1 + LambertW(1)) * LambertW(1)^(n + 1)). - Vaclav Kotesovec, Mar 08 2018
Or: a(n) ~ (1/sqrt(1+w)) * exp(1/w-2)^n * (n/w)^(n+1), with w = LambertW(1) ~ 0.56714329... The relative error decreases from 10^-2 for a(2) to 10^-3 for a(15), but reaches 10^-3.5 only at a(45). - M. F. Hasler, Mar 09 2018

A335975 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1) + x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 15, 1, 1, 5, 19, 47, 52, 1, 1, 6, 29, 103, 227, 203, 1, 1, 7, 41, 189, 622, 1215, 877, 1, 1, 8, 55, 311, 1357, 4117, 7107, 4140, 1, 1, 9, 71, 475, 2576, 10589, 29521, 44959, 21147, 1, 1, 10, 89, 687, 4447, 23031, 88909, 227290, 305091, 115975, 1

Offset: 0

Seiichi Manyama, Jul 03 2020

			Square array begins:
  1,   1,    1,     1,     1,      1,      1, ...
  1,   2,    3,     4,     5,      6,      7, ...
  1,   5,   11,    19,    29,     41,     55, ...
  1,  15,   47,   103,   189,    311,    475, ...
  1,  52,  227,   622,  1357,   2576,   4447, ...
  1, 203, 1215,  4117, 10589,  23031,  44683, ...
  1, 877, 7107, 29521, 88909, 220341, 478207, ...

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

Columns k=0-4 give: A000012, A000110(n+1), A035009(n+1), A078940, A078945.
Main diagonal gives A334240.
Cf. A292860, A335977.

T[0, k_] := 1; T[n_, k_] := T[n - 1, k] + k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)

T(0,k) = 1 and T(n,k) = T(n-1,k) + k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

T(n,k) = exp(-k) * Sum_{j>=0} (j + 1)^n * k^j / j!.

A346654 a(n) = Bell(2*n,n).

Original entry on oeis.org

1, 2, 94, 12351, 3188340, 1362057155, 869725707522, 775929767223352, 921839901090823112, 1406921223577401454239, 2682502220690005671884710, 6248503930824315386034050253, 17460431497766377837983159782652, 57647207262184459310081410522242310, 222006095854149044448961838142906736554

Offset: 0

Vaclav Kotesovec, Jul 27 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..198

Cf. A000110, A189233, A242817, A292860, A346655.

b:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..14);  # Alois P. Heinz, Jul 27 2021

Mathematica
```
Table[BellB[2*n, n], {n, 0, 20}]
```

a(n) ~ 4^n * exp((2/LambertW(2) - 3)*n) * n^(2*n) / (sqrt(1 + LambertW(2)) * LambertW(2)^(2*n)).

a(n) = A189233(2n,n) = A292860(2n,n). - Alois P. Heinz, Jul 27 2021

A346655 a(n) = Bell(3*n,n).

Original entry on oeis.org

1, 5, 2430, 5597643, 35618229364, 483040313859705, 11977437107679230274, 490630568583958198181583, 30889771581097736768046865352, 2832037863467651034046820871428061, 362579939205426756198837321528946171110, 62687814132880422794200073791149602981717667

Offset: 0

Vaclav Kotesovec, Jul 27 2021

nonn

In general, for k>=1, Bell(k*n,n) ~ (k*n/LambertW(k))^(k*n) / (sqrt(1 + LambertW(k)) * exp(n*(k + 1 - k/LambertW(k)))).

Alois P. Heinz, Table of n, a(n) for n = 0..137

Cf. A000110, A189233, A242817, A292860, A346654.

b:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
    end:
a:= n-> b(3*n, n):
seq(a(n), n=0..11);  # Alois P. Heinz, Jul 27 2021

Mathematica
```
Table[BellB[3*n, n], {n, 0, 15}]
```

a(n) ~ (3*n/LambertW(3))^(3*n) / (sqrt(1 + LambertW(3)) * exp(n*(4 - 3/LambertW(3)))).

a(n) = A189233(3n,n) = A292860(3n,n). - Alois P. Heinz, Jul 27 2021

A343263 a(0) = 1; a(n+1) = exp(-a(n)) * Sum_{k>=0} a(n)^k * k^n / k!.

Original entry on oeis.org

1, 1, 1, 2, 22, 301554, 2493675105669492542968967478

Offset: 0

Ilya Gutkovskiy, Apr 09 2021

nonn

The next term is too large to include.

Cf. A000110, A003659, A067691, A110386, A189233, A242817, A292860, A294373.

a[0] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 1, k] a[n - 1]^k, {k, 0, n - 1}]; Table[a[n], {n, 0, 6}]
a[0] = 1; a[n_] := a[n] = BellB[n - 1, a[n - 1]]; Table[a[n], {n, 0, 6}]

a(0) = 1; a(n+1) = n! * [x^n] exp(a(n) * (exp(x) - 1)).

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

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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A357681 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * (exp(x) - 1) ).

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A350256 Triangle read by rows. T(n, k) = BellPolynomial(n, k).

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A351761 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.

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A299824 a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)!.

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A335975 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1) + x).

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A346654 a(n) = Bell(2*n,n).

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A299824 a(n) = (1/e^n)Sum_{j >= 1} j^n n^j / (j-1)!.