A323771 Expansion of e.g.f. 2*exp(x/(2 - 2*x))*sinh(sqrt(5)*x/(2 - 2*x))/sqrt(5).

0, 1, 3, 14, 87, 665, 5978, 61459, 709037, 9053386, 126595315, 1922334679, 31480716312, 552776980001, 10356230986023, 206133285278530, 4342815027527307, 96526112076314221, 2256839592693577138, 55361051241071952659, 1421458419738657242545, 38121104146852228186886

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000045(k)*n!/k!.

From Vaclav Kotesovec, Jan 27 2019: (Start)

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).

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)

Mathematica

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]!]
Table[Sum[Binomial[n - 1, k - 1] Fibonacci[k] n!/k!, {k, 0, n}], {n, 0, 21}]

Crossrefs

Cf. A000045.

Sequence in context: A308878 A051818 A091102 * A325140 A132624 A121587

Adjacent sequences: A323768 A323769 A323770 * A323772 A323773 A323774

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 27 2019

Status

approved