A355495 -id:A355495 - cp's OEIS frontend

A355494 Expansion of Sum_{k>=0} (k * x/(1 - x))^k.

1, 1, 5, 36, 350, 4328, 65132, 1155904, 23640724, 547544032, 14166236708, 404944248104, 12674392793900, 431104742439088, 15834117059443828, 624575921756875960, 26332801242942780668, 1181750740315156943936, 56244454481507648435012

Offset: 0

Seiichi Manyama, Jul 04 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..385

Cf. A355495, A355496.

Cf. A086331.

Join[{1}, Table[Sum[k^k * Binomial[n-1,k-1], {k,1,n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jul 05 2022 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x/(1-x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k^k*binomial(n-1, k-1)));

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

a(n) ~ exp(exp(-1)) * n^n. - Vaclav Kotesovec, Jul 05 2022

A355496 Expansion of Sum_{k>=0} (k^3 * x/(1 - x))^k.

Original entry on oeis.org

1, 1, 65, 19812, 16836458, 30584805344, 101712712528352, 559155681922806328, 4726278437746021089208, 58187531579876705928027712, 1000523517685151396828602120640, 23235157037192774575979788565151104, 709057406693306876515431403267191583808

Offset: 0

Seiichi Manyama, Jul 04 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..152

Cf. A355494, A355495.

Cf. A355472, A355493.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^3*x/(1-x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k^(3*k)*binomial(n-1, k-1)));

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

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A355494 Expansion of Sum_{k>=0} (k * x/(1 - x))^k.

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A355496 Expansion of Sum_{k>=0} (k^3 * x/(1 - x))^k.

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PARI

PARI

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