A336188 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^n.

1, 2, 13, 352, 38401, 16971876, 29359436149, 207003074670848, 5679112509686022145, 636468045901197095750500, 277939985126193076692203962501, 494649880078824954885176565423811200, 3447375085398645453825889951638344722092289, 97424105704407389799712313421357308088296084669504

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..57

Vaclav Kotesovec, Plot of a(n) / (2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2)) for n = 1..10000000

Formula

Let f(n) = 2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2). For sufficiently large n 0.7675... < a(n)/f(n) < 0.7900... - Vaclav Kotesovec, Jul 11 2020

The above bounds of Vaclav Kotesovec can be recast as: |a(n)/f(n) - exp(-1/4)| <= (3*Pi)^(-2) for sufficiently large n. - Peter Luschny, Jul 12 2020

a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-2), exp(-4)) * 2^(n^2 + n/2) / Pi^(n/2) if n is even and a(n) ~ exp(-3/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-4), exp(-4)) * 2^(n^2 + n/2) * sqrt(n) / Pi^(n/2) if n is odd. - Vaclav Kotesovec, Jul 13 2020

Mathematica

Unprotect[Power]; 0^0 = 1; a[n_] := Sum[n^k * Binomial[n, k]^n, {k, 0, n} ]; Array[a, 14, 0] (* Amiram Eldar, Jul 11 2020 *)

Prog

(PARI) {a(n) = sum(k=0, n, n^k*binomial(n, k)^n)}
(Magma) [(&+[n^j*Binomial(n, j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022
(SageMath) [sum(n^j*binomial(n, j)^n for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Crossrefs

Main diagonal of A336187.

Cf. A167010, A187021, A234971, A241247, A336202, A336213, A336214.

Sequence in context: A304727 A355730 A075620 * A348015 A131525 A366722

Adjacent sequences: A336185 A336186 A336187 * A336189 A336190 A336191

Keyword

nonn,nice

Author

Seiichi Manyama, Jul 11 2020

Status

approved