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Expansion of e.g.f. 1/(1 - x - x^2)^(x^2).
3

%I #12 Mar 14 2024 08:49:51

%S 1,0,0,6,36,160,1620,18648,220080,2924640,44775360,753207840,

%T 13836731040,276442882560,5972081379264,138607594171200,

%U 3440465206214400,90951997553464320,2551374460670538240,75694365919478960640,2368107785432883916800

%N Expansion of e.g.f. 1/(1 - x - x^2)^(x^2).

%F a(n) = n! * Sum_{j=0..n} Sum_{k=0..floor(j/2)} binomial(j-k,n-j-k) * |Stirling1(j-k,k)|/(j-k)!.

%F a(n) ~ sqrt(2*Pi) * phi^(n + 1/phi^2) * n^(n + 3/2 - phi)/ (Gamma(1/phi^2) * 5^(1/(2*phi^2)) * exp(n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Mar 14 2024

%o (PARI) a(n) = n!*sum(j=0, n, sum(k=0, j\2, binomial(j-k, n-j-k)*abs(stirling(j-k, k, 1))/(j-k)!));

%Y Cf. A088369, A371158.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Mar 13 2024