A351733 -id:A351733 - 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)

A351734 Expansion of e.g.f. exp( 3 * x * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 6, 9, 120, 555, 5148, 39711, 378528, 3715011, 39838260, 452684463, 5463506304, 69553644771, 930940368036, 13054086036855, 191222363275968, 2918620069099395, 46309955947643124, 762335523354333855, 12995722456718984160, 229045407317491457763

Offset: 0

Seiichi Manyama, May 20 2022

nonn

Cf. A052506, A351733.

Cf. A053490, A351737.

With[{nn=30},CoefficientList[Series[Exp[3x (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 02 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(3*x*(exp(x)-1))))

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

a(n) = n! * Sum_{k=0..floor(n/2)} 3^k * Stirling2(n-k,k)/(n-k)!.

A354311 Expansion of e.g.f. exp( x/2 * (exp(2 * x) - 1) ).

Original entry on oeis.org

1, 0, 2, 6, 28, 160, 1056, 7784, 63568, 569664, 5542240, 58038112, 650045760, 7746901760, 97790608384, 1302349549440, 18235836899584, 267663541270528, 4107395264113152, 65739857693144576, 1095095457262013440, 18949711553467957248, 340036076121127395328

Offset: 0

Seiichi Manyama, May 23 2022

sign

Cf. A052506, A354312.

Cf. A351733, A351736, A354309, A354313.

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

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

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

a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-2) * binomial(n-1,k-1) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * Stirling2(n-k,k)/(n-k)!.

A361652 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^j * Stirling2(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, 6, 16, 0, 1, 0, 8, 9, 56, 65, 0, 1, 0, 10, 12, 120, 250, 336, 0, 1, 0, 12, 15, 208, 555, 1812, 1897, 0, 1, 0, 14, 18, 320, 980, 5148, 12614, 11824, 0, 1, 0, 16, 21, 456, 1525, 11064, 39711, 101040, 80145, 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,   6,   9,  12,   15, ...
  0, 16,  56, 120, 208,  320, ...
  0, 65, 250, 555, 980, 1525, ...

Columns k=0..3 give: A000007, A052506, A351733, A351734.
Main diagonal gives (-1)^n * A290158(n).
Cf. A362834.

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

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

cp's OEIS Frontend

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

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A351734 Expansion of e.g.f. exp( 3 * x * (exp(x) - 1) ).

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A354311 Expansion of e.g.f. exp( x/2 * (exp(2 * x) - 1) ).

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A361652 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^j * Stirling2(n-j,j)/(n-j)!.

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