A336213 a(n) = Sum_{k=0..n} k^k * binomial(n,k)^n, with a(0)=1.

1, 2, 9, 163, 12609, 3906251, 4835455813, 23882051929709, 470073929716006913, 36867039626275056203923, 11562789460238169439667262501, 14393917436542502296957220221339601, 72060131612303615870363237649174605005057, 1424448870088911493303605765206905153730451241313

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..58

Vaclav Kotesovec, Plot of a(n)/f(n) for n = 1..10000000

Formula

Let f(n) = exp(-1/4) * QPochhammer(exp(-4)) * 2^(n^2 - 1/4) * exp((3*log(n)^2 + 3*log(2)^2 + Pi^2 - 1)/24) * n^((1 - log(2))/4) / Pi^(n/2). For sufficiently large n 0.985... < a(n)/f(n) < 1.015...

a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-1)/2, exp(-4)) * 2^(n^2) / Pi^(n/2) if n is even and a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-3)/2, exp(-4)) * sqrt(n) * 2^(n^2 - 1/2) / Pi^(n/2) if n is odd.

Mathematica

Table[1 + Sum[k^k * Binomial[n, k]^n, {k, 1, n}], {n, 0, 15}]

Prog

(PARI) a(n) = if (n==0, 1, sum(k=0, n, k^k * binomial(n, k)^n)); \\ Michel Marcus, Jul 13 2020

Crossrefs

Cf. A072034, A086331, A167008, A167010, A336188, A336204, A336212, A336214.

Sequence in context: A133468 A182948 A081459 * A038843 A367526 A237201

Adjacent sequences: A336210 A336211 A336212 * A336214 A336215 A336216

Keyword

nonn

Author

Vaclav Kotesovec, Jul 12 2020

Status

approved