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%I #44 Apr 30 2021 01:08:04
%S 1,1,2,9,83,1268,62283,10296321,2668655428,1306416217435,
%T 3055324257386077,17213278350960504924,137320554100797006975445,
%U 3087543920644806918694851647,335732238884967561227813578781572,61125387696211835948801235842204794881
%N a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n.
%C 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
%H Seiichi Manyama, <a href="/A323769/b323769.txt">Table of n, a(n) for n = 0..71</a>
%H Vaclav Kotesovec, <a href="/A323769/a323769.jpg">Graph - the asymptotic ratio</a>
%H Vaclav Kotesovec, <a href="/A323769/a323769_1.jpg">Graph - dependence of the limit on the fractional part of n/(sqrt(5)*phi)</a>
%F 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
%F 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
%t Table[Sum[Binomial[n-k,k]^n, {k, 0, n/2}], {n, 0, 15}] (* _Vaclav Kotesovec_, Jan 27 2019 *)
%o (PARI) {a(n) = sum(k=0, n\2, binomial(n-k, k)^n)}
%Y Main diagonal of A323767.
%Y Cf. A011973, A051286, A181545, A181546, A181547, A209428, A323768.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jan 27 2019
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