A336204 a(n) = Sum_{k=0..n} 2^k * binomial(n,k)^n.

1, 3, 13, 171, 7761, 1256283, 741398869, 1609036666443, 13118066779885825, 399221556627301207443, 46476897754761801245056293, 20377119057713827002258336842283, 34592895120825704155462768381947657489, 222457046333769635263635086646525921070978443

Offset: 0

Author

Seiichi Manyama, Jul 11 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..59

Crossrefs

Main diagonal of A336203.

Cf. A167010, A336188, A336202, A336212.

Programs

Magma

[(&+[2^j*Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022

Mathematica

Table[Sum[2^k*Binomial[n, k]^n, {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 12 2020 *)

PARI

{a(n) = sum(k=0, n, 2^k*binomial(n, k)^n)};

SageMath

def A336204(n): return sum(2^k*binomial(n,k)^n for k in (0..n))
[A336204(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

Formula

a(n) ~ c * 2^(n*(n+1)) / (Pi*n)^(n/2), where c = exp(-1/4) * Sum_{k = -oo..oo} 2^k * exp(-2*k^2) = 1.0434092897163574491113380912895917... if n is even and c = exp(-1/4) * Sum_{k = -oo..oo} 2^(k + 1/2) * exp(-2*(k + 1/2)^2) = 1.029587234777114329090639723058125257... if n is odd. - Vaclav Kotesovec, Jul 12 2020