A360018 -id:A360018 - cp's OEIS frontend

A360775 Expansion of Sum_{k>=0} (x * (k + x^2))^k.

1, 1, 4, 28, 260, 3152, 46913, 826677, 16823968, 388245283, 10016796672, 285699444297, 8926107792609, 303160590533808, 11120927427841820, 438196895219227683, 18457860168281435172, 827678295600605015006, 39364859979651634985089

Offset: 0

Seiichi Manyama, Feb 20 2023

nonn

Cf. A360774, A360776.

Cf. A360018, A360727.

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

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

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

cp's OEIS Frontend

A360775 Expansion of Sum_{k>=0} (x * (k + x^2))^k.

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

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