A360618 Expansion of Sum_{k>=0} (k * x * (1 + k*x))^k.

1, 1, 5, 43, 515, 7950, 150086, 3349945, 86296849, 2519907605, 82249222661, 2967449372028, 117266100841668, 5037282382077353, 233701540415817409, 11645959855678136519, 620389246928233860127, 35181554115178393462386, 2116059351692554708911298

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..373

Formula

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

a(n) ~ c * d^n * n^n, where d = (1-r)^(2-r) / (r^r * (1-2*r)^(1-2*r)) where r = 0.163662210494891118101893756356803907477984542... is the root of the equation (1-2*r)^2 = r*(1-r) * exp(1/(1-r)) and c = 0.78619174295244329885973980954744130517052330684023764340463604028671858569... - Vaclav Kotesovec, Feb 14 2023

Mathematica

Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^n, {k, 0, n/2}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 14 2023 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+k*x))^k))
(PARI) a(n) = sum(k=0, n\2, (n-k)^n*binomial(n-k, k));

Crossrefs

Cf. A072034, A360592, A360611.

Sequence in context: A107720 A362188 A299425 * A060053 A227176 A132691

Adjacent sequences: A360615 A360616 A360617 * A360619 A360620 A360621

Keyword

nonn

Author

Seiichi Manyama, Feb 14 2023

Status

approved