A355496 -id:A355496 - 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

A355495 Expansion of Sum_{k>=0} (k^2 * x/(1 - x))^k.

Original entry on oeis.org

1, 1, 17, 762, 67772, 10032208, 2226273192, 691431572992, 286268594755712, 152365547943819264, 101361042063083269520, 82409537565402784477984, 80397802305461995791664944, 92692687015689239272783171264

Offset: 0

Seiichi Manyama, Jul 04 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..213

Cf. A355494, A355496.

Cf. A323280.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x/(1-x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k^(2*k)*binomial(n-1, k-1)));

a(n) = Sum_{k=1..n} k^(2*k) * 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).

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

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A355495 Expansion of Sum_{k>=0} (k^2 * x/(1 - x))^k.

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

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