A294499 -id:A294499 - cp's OEIS frontend

A266497 Binomial transform of A015128.

1, 3, 9, 27, 79, 225, 627, 1717, 4633, 12341, 32501, 84737, 218959, 561263, 1428287, 3610671, 9072367, 22668285, 56345835, 139382713, 343242533, 841713531, 2055944117, 5003148987, 12132552115, 29323810757, 70651867863, 169719163521, 406541986857, 971192810019

Offset: 0

Vaclav Kotesovec, Dec 30 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A015128, A218481, A266232, A294499.

A015128[n_]:=Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}];
Table[Sum[Binomial[n, k]*A015128[k], {k, 0, n}], {n, 0, 30}]

a(n) ~ 2^(n-2) * exp(Pi*sqrt(n/2) + Pi^2/16) / n.

a(n) = [x^n] (1 + x)^n/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018

A307945 Exponential convolution of A015128 with themselves.

Original entry on oeis.org

1, 4, 16, 64, 252, 968, 3616, 13120, 46432, 160772, 545856, 1821056, 5979520, 19350552, 61795968, 194964672, 608261628, 1878140024, 5743681784, 17408223328, 52320105080, 156011658272, 461763417056, 1357182242560, 3962591708576, 11497241014652

Offset: 0

Vaclav Kotesovec, May 07 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..2994

Cf. A015128, A266497, A294499, A307755, A307756.

S:= series(1/JacobiTheta4(0,q),q,101):
f:= n -> add(binomial(n,k)*coeff(S,q,k)*coeff(S,q,n-k),k=0..n):
map(f, [$0..100]); # Robert Israel, May 08 2019

A015128[n_]:=Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}]; Table[Sum[Binomial[n, k]*A015128[k]*A015128[n-k], {k, 0, n}], {n, 0, 25}]

a(n) = Sum_{k=0..n} binomial(n,k) * A015128(k) * A015128(n-k).

a(n) ~ 2^(n-4) * exp(Pi*sqrt(2*n)) / n^2.

cp's OEIS Frontend

A266497 Binomial transform of A015128.

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