A355470 -id:A355470 - cp's OEIS frontend

A355466 Expansion of Sum_{k>=0} (k^k * x)^k/(1 - k^k * x)^(k+1).

1, 2, 19, 19879, 4297094601, 298028721578591321, 10314430386430205371442173873, 256923580889667562995278943476559835493321, 6277101737079381674883855772624745947410338680458857322625

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A072034, A242446, A355470.

Cf. A349886.

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, (k^k*x)^k/(1-k^k*x)^(k+1)))

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(k^k*x)*(k^k*x)^k/k!)))

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

E.g.f.: Sum_{k>=0} exp(k^k * x) * (k^k * x)^k/k!.

a(n) = Sum_{k=0..n} k^(k*n) * binomial(n,k).

A355469 Expansion of Sum_{k>=0} (k^3 * x/(1 - k^3 * x))^k.

Original entry on oeis.org

1, 1, 65, 20708, 18383828, 34898769936, 121324513279512, 697408243146701056, 6165037130760825320768, 79390334273383043609851520, 1428007543233019703635181454080, 34693490969752778534655707874499584, 1107666867764009444258160579726602423808

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..151

Cf. A355468, A355470.

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

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

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

A355493 Expansion of Sum_{k>=0} (k^3 * x)^k/(1 - x)^(k+1).

Original entry on oeis.org

1, 2, 67, 19879, 16856337, 30601661681, 101743314190033, 559257425236996361, 4726837695171258085569, 58192258417571877186113281, 1000581709943568968705788233921, 23236157618902718144948494353385025, 709080642850925779233576351761544968833

Offset: 0

Seiichi Manyama, Jul 04 2022

nonn

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

Cf. A086331, A323280.

Cf. A355470, A355473, A355496.

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

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

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

E.g.f.: exp(x) * Sum_{k>=0} (k^3 * x)^k/k!.

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

cp's OEIS Frontend

A355466 Expansion of Sum_{k>=0} (k^k * x)^k/(1 - k^k * x)^(k+1).

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A355469 Expansion of Sum_{k>=0} (k^3 * x/(1 - k^3 * x))^k.

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A355493 Expansion of Sum_{k>=0} (k^3 * x)^k/(1 - x)^(k+1).

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