A323769 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n.

1, 1, 2, 9, 83, 1268, 62283, 10296321, 2668655428, 1306416217435, 3055324257386077, 17213278350960504924, 137320554100797006975445, 3087543920644806918694851647, 335732238884967561227813578781572, 61125387696211835948801235842204794881

Offset: 0

Author

Seiichi Manyama, Jan 27 2019

Keywords

• nonn

Comments

The limit a(n) / (5^(n/4) * phi^(n*(n+1)) / (2*Pi*n)^(n/2)) does not exist but oscillates between 2 attractors. The value is dependent on the fractional part of n/(sqrt(5)*phi), see graph. - Vaclav Kotesovec, Jan 28 2019

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..71
• Vaclav Kotesovec, Graph - the asymptotic ratio
• Vaclav Kotesovec, Graph - dependence of the limit on the fractional part of n/(sqrt(5)*phi)

Crossrefs

Main diagonal of A323767.

Cf. A011973, A051286, A181545, A181546, A181547, A209428, A323768.

Programs

Mathematica

Table[Sum[Binomial[n-k,k]^n, {k, 0, n/2}], {n, 0, 15}] (* Vaclav Kotesovec, Jan 27 2019 *)

PARI

{a(n) = sum(k=0, n\2, binomial(n-k, k)^n)}

Formula

a(n)^(1/n) ~ 5^(1/4) * phi^(n+1) / sqrt(2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 27 2019

log(a(n)) ~ n*(n*v + w - log(n))/2 with v = 2*log((1 + sqrt(5))/2) and w = log((35 + 15*sqrt(5))/(8*Pi^2))/2, preceding formula recast. - Peter Luschny, Jan 28 2019