The OEIS is supported by the many generous donors to the OEIS Foundation.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #11 Jan 30 2020 21:29:18
%S 0,1,3,14,87,665,5978,61459,709037,9053386,126595315,1922334679,
%T 31480716312,552776980001,10356230986023,206133285278530,
%U 4342815027527307,96526112076314221,2256839592693577138,55361051241071952659,1421458419738657242545,38121104146852228186886
%N Expansion of e.g.f. 2*exp(x/(2 - 2*x))*sinh(sqrt(5)*x/(2 - 2*x))/sqrt(5).
%F a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000045(k)*n!/k!.
%F From _Vaclav Kotesovec_, Jan 27 2019: (Start)
%F D-finite with recurrence: a(n) = (4*n - 5)*a(n-1) - (6*n^2 - 22*n + 19)*a(n-2) + (n-3)*(n-2)*(4*n - 9)*a(n-3) - (n-4)*(n-3)^2*(n-2)*a(n-4).
%F a(n) ~ phi^(1/4) * n^(n - 1/4) / (sqrt(10) * exp(n - 2*sqrt(phi*n) + phi/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. (End)
%t FullSimplify[nmax = 21; CoefficientList[Series[2 Exp[x/(2 - 2 x)] Sinh[Sqrt[5] x/(2 - 2 x)]/Sqrt[5], {x, 0, nmax}], x] Range[0, nmax]!]
%t Table[Sum[Binomial[n - 1, k - 1] Fibonacci[k] n!/k!, {k, 0, n}], {n, 0, 21}]
%Y Cf. A000045.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 27 2019