A351734 -id:A351734 - cp's OEIS frontend

A351737 Expansion of e.g.f. exp( x * (exp(3 * x) - 1) ).

1, 0, 6, 27, 216, 2025, 21708, 260253, 3460320, 50395041, 795324420, 13495904829, 244747554912, 4718754452529, 96285948702804, 2071265238290565, 46815054101658432, 1108489016781839169, 27424412680091114628, 707277138662880504045, 18974871706141125008640

Offset: 0

Seiichi Manyama, May 20 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..461

Cf. A052506, A351736.

Cf. A351734, A351735.

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

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

a(n) = sum(k=0, n, (3*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-(3*k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * Stirling2(n-k,k)/(n-k)!.
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (3*k-1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} x^k / (1 - (3*k-1)*x)^(k+1). (End)

A351733 Expansion of e.g.f. exp( 2 * x * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 4, 6, 56, 250, 1812, 12614, 101040, 864882, 7988780, 78726142, 823897032, 9111774698, 106068603396, 1295153135670, 16538681229152, 220281968528098, 3053087839536732, 43941561067048430, 655501502129291640, 10118103843683127642

Offset: 0

Seiichi Manyama, May 20 2022

nonn

Cf. A052506, A351734.

Cf. A053489, A351736.

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

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

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

A354312 Expansion of e.g.f. exp( x/3 * (exp(3 * x) - 1) ).

Original entry on oeis.org

1, 0, 2, 9, 48, 315, 2496, 22491, 223728, 2437371, 28931040, 371291283, 5111412120, 75014135235, 1168157451384, 19228202401635, 333378840718944, 6069073767712587, 115683487658404272, 2303091818149762899, 47784447190060311240, 1031179733234906055507

Offset: 0

Seiichi Manyama, May 23 2022

sign

Cf. A052506, A354311.

Cf. A351734, A351737, A354310, A354314.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/3*(exp(3*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*3^(j-2)*binomial(i-1, j-1)*v[i-j+1])); v;

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

a(0) = 1; a(n) = Sum_{k=2..n} k * 3^(k-2) * binomial(n-1,k-1) * a(n-k).

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

A351737 Expansion of e.g.f. exp( x * (exp(3 * x) - 1) ).

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A351733 Expansion of e.g.f. exp( 2 * x * (exp(x) - 1) ).

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A354312 Expansion of e.g.f. exp( x/3 * (exp(3 * 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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