A053491 -id:A053491 - cp's OEIS frontend

A351736 Expansion of e.g.f. exp( x * (exp(2 * x) - 1) ).

1, 0, 4, 12, 80, 560, 4512, 40768, 407808, 4453632, 52605440, 667234304, 9032423424, 129822564352, 1972450443264, 31559866736640, 530043925495808, 9317136303718400, 170976603113127936, 3268020569256755200, 64928967058257346560, 1338431135849666052096

Offset: 0

Seiichi Manyama, May 20 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..488

Cf. A052506, A351737.

Cf. A053491, A351733.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(2*x)-1))))

a(n) = n!*sum(k=0, n\2, 2^(n-k)*stirling(n-k, k, 2)/(n-k)!);

a(n) = sum(k=0, n, (2*k-1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(2*k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-k) * Stirling2(n-k,k)/(n-k)!.
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (2*k-1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} x^k / (1 - (2*k-1)*x)^(k+1). (End)

A354309 Expansion of e.g.f. 1/(1 - 2*x)^(x/2).

Original entry on oeis.org

1, 0, 2, 6, 44, 360, 3744, 46200, 662864, 10838016, 198943200, 4050937440, 90613710912, 2208677328000, 58265734055424, 1653914478303360, 50263564166365440, 1628300694034022400, 56012708047907510784, 2039053421375533094400, 78314004507947110456320

Offset: 0

Seiichi Manyama, May 23 2022

nonn

Cf. A066166, A354310.

Cf. A053491, A354311, A354315, A354319.

With[{nn=20},CoefficientList[Series[1/(1-2x)^(x/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 10 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x)^(x/2)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, i, j*2^(j-2)/(j-1)*v[i-j+1]/(i-j)!)); v;

a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*abs(stirling(n-k, k, 1))/(n-k)!);

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * 2^(k-2)/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * |Stirling1(n-k,k)|/(n-k)!.
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n)). - Vaclav Kotesovec, Mar 14 2024

A351735 Expansion of e.g.f. 1/(1 - 3 * x)^x.

Original entry on oeis.org

1, 0, 6, 27, 324, 4050, 64962, 1224720, 26776656, 665390376, 18529576200, 571602980520, 19349098690248, 713092338026640, 28422424039182768, 1218246132898693080, 55876497505161154560, 2730710993688989519040, 141654212475516155694528

Offset: 0

Seiichi Manyama, May 20 2022

Cf. A053490, A053491.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x)^x))

a(n) = n!*sum(k=0, n\2, 3^(n-k)*abs(stirling(n-k, k, 1))/(n-k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * |Stirling1(n-k,k)|/(n-k)!.

a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 1/6) / (Gamma(1/3) * exp(n)). - Vaclav Kotesovec, Aug 30 2025

A362837 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling1(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 20, 0, 1, 0, 8, 27, 112, 90, 0, 1, 0, 10, 48, 324, 960, 594, 0, 1, 0, 12, 75, 704, 4050, 10848, 4200, 0, 1, 0, 14, 108, 1300, 11520, 64962, 141120, 34544, 0, 1, 0, 16, 147, 2160, 26250, 239616, 1224720, 2122496, 316008, 0

Offset: 0

Seiichi Manyama, May 05 2023

			Square array begins:
  1,  1,   1,    1,     1,     1, ...
  0,  0,   0,    0,     0,     0, ...
  0,  2,   4,    6,     8,    10, ...
  0,  3,  12,   27,    48,    75, ...
  0, 20, 112,  324,   704,  1300, ...
  0, 90, 960, 4050, 11520, 26250, ...

Columns k=0..3 give: A000007, A066166, A053491, A351735.
Main diagonal gives A362838.
Cf. A362834.

T(n, k) = (-1)^n*n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 1)/(n-j)!);

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

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A351736 Expansion of e.g.f. exp( x * (exp(2 * x) - 1) ).

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A354309 Expansion of e.g.f. 1/(1 - 2*x)^(x/2).

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A351735 Expansion of e.g.f. 1/(1 - 3 * x)^x.

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A362837 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling1(n-j,j)/(n-j)!.

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