A360032 -id:A360032 - cp's OEIS frontend

A360776 Expansion of Sum_{k>=0} (x * (k + x^3))^k.

1, 1, 4, 27, 257, 3129, 46683, 823799, 16780342, 387467154, 10000823639, 285328449077, 8916487888186, 302885106945216, 11112292144568909, 437902806653498835, 18447046953316227905, 827251374022851280231, 39346845973273509115167

Offset: 0

Seiichi Manyama, Feb 20 2023

nonn

Cf. A360774, A360775.

Cf. A360032, A360728.

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

a(n) = sum(k=0, n\4, (n-3*k)^(n-4*k)*binomial(n-3*k, k));

a(n) = Sum_{k=0..floor(n/4)} (n-3*k)^(n-4*k) * binomial(n-3*k,k).

A360018 Expansion of Sum_{k>=0} (k * x * (1 + (k * x)^2))^k.

Original entry on oeis.org

1, 1, 4, 28, 288, 3854, 63104, 1220729, 27248128, 689446671, 19501121536, 609753349945, 20883798220800, 777529328875208, 31266494467227648, 1350520199148276667, 62360172065142341632, 3065369553470816704832, 159818389764050045894656

Offset: 0

Seiichi Manyama, Feb 19 2023

nonn

Cf. A360032, A360618.

Cf. A360730.

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

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

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

a(n) ~ c * (1-2*r)^(2*(1-r)*n) * n^n / ((1-3*r)^((1-3*r)*n) * r^(r*n)), where r = 0.06730326916452804898090832100482072129668759014637687455288... is the root of the equation (1-2*r) * log((1-3*r)^3 / (r*(1-2*r)^2)) = 2 and c = 0.77456580764856204420602709595934338976380573814558378938814706465915... - Vaclav Kotesovec, Feb 20 2023

cp's OEIS Frontend

A360776 Expansion of Sum_{k>=0} (x * (k + x^3))^k.

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A360018 Expansion of Sum_{k>=0} (k * x * (1 + (k * x)^2))^k.

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