A360835 -id:A360835 - cp's OEIS frontend

A360834 Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^2)^(k+1).

1, 1, 4, 29, 304, 4100, 67520, 1314167, 29520128, 751658635, 21393444864, 673046604600, 23192501108736, 868730852002205, 35145114836811776, 1527192185786650417, 70941146068492943360, 3508043437942077557884, 183989995827118805352448

Offset: 0

Seiichi Manyama, Feb 22 2023

nonn

Cf. A072034, A360835.

Cf. A000045, A360782, A360787.

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

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

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

A360833 Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^3) )^k.

Original entry on oeis.org

1, 1, 4, 27, 257, 3189, 48843, 889080, 18731109, 448004763, 11987812504, 354763577414, 11503684020051, 405589341060930, 15447798292502206, 632069580794524857, 27649951709582591394, 1287748889361331630661, 63616184683123273364961

Offset: 0

Seiichi Manyama, Feb 22 2023

nonn

Cf. A195242, A360832.

Cf. A338661, A360811, A360835.

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

a(n) = sum(k=0, n\3, (n-3*k)^n*binomial(n-2*k-1, k));

a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^n * binomial(n-2*k-1,k).

cp's OEIS Frontend

A360834 Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^2)^(k+1).

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A360833 Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^3) )^k.

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Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula