A360731 -id:A360731 - cp's OEIS frontend

A360730 Expansion of Sum_{k>=0} (k * x * (1 + k*x^2))^k.

1, 1, 4, 28, 272, 3368, 50768, 902397, 18481408, 428556075, 11099001600, 317544062217, 9946366838784, 338537433281448, 12441407233436672, 491002325860132371, 20710640842719301632, 929821866165431838038, 44270378887441746923520

Offset: 0

Seiichi Manyama, Feb 18 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..385

Cf. A360618, A360731.

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-k)*binomial(n-2*k, k));

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

A360728 Expansion of Sum_{k>=0} (k * x * (1 + x^3))^k.

Original entry on oeis.org

1, 1, 4, 27, 257, 3133, 46737, 824567, 16792845, 387700506, 10005766337, 285445919589, 8919587932524, 302975123887680, 11115145723728035, 438000897534309171, 18450681900124075166, 827395845674975999727, 39352977072147424071861

Offset: 0

Seiichi Manyama, Feb 18 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..385

Cf. A360611, A360727.

Cf. A360731.

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

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

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

A360032 Expansion of Sum_{k>=0} (k * x * (1 + (k * x)^3))^k.

Original entry on oeis.org

1, 1, 4, 27, 257, 3189, 48843, 889079, 18730597, 447945714, 11983618199, 354519428597, 11490618543066, 404910044246256, 15412461332440829, 630199633730994675, 27548323149955792880, 1282044807268698303751, 63284535745130267484867

Offset: 0

Seiichi Manyama, Feb 19 2023

nonn

Cf. A360018, A360618.

Cf. A360731.

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\4, (n-3*k)^n*binomial(n-3*k, k));

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

A360749 Expansion of Sum_{k>=0} (x * (1 + k*x^3))^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 10, 17, 30, 64, 146, 315, 649, 1386, 3164, 7381, 16931, 38649, 90406, 217474, 527586, 1277452, 3112371, 7705059, 19336789, 48800634, 123617380, 315671212, 814711955, 2119996540, 5545342621, 14584694613, 38641783669, 103158314515

Offset: 0

Seiichi Manyama, Feb 19 2023

nonn

Cf. A360592, A360748.

Cf. A003269, A360731, A360747.

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+k*x^3))^k))

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

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

cp's OEIS Frontend

A360730 Expansion of Sum_{k>=0} (k * x * (1 + k*x^2))^k.

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A360728 Expansion of Sum_{k>=0} (k * x * (1 + x^3))^k.

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A360032 Expansion of Sum_{k>=0} (k * x * (1 + (k * x)^3))^k.

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A360749 Expansion of Sum_{k>=0} (x * (1 + k*x^3))^k.

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