A394993 a(n) = n! * Sum_{k=0..n} (-1)^(k+1)*F(k)/k!, where F(n) is the n-th Fibonacci number, A000045(n).

0, 1, 1, 5, 17, 90, 532, 3737, 29875, 268909, 2689035, 29579474, 354953544, 4614396305, 64601547893, 969023219005, 15504371503093, 263574315554178, 4744337679972620, 90142415919483961, 1802848318389672455, 37859814686183132501, 832915923096028897311

Offset

0,4

Comments

This sequence occurs in the setting of expressing Euler's number e as the ratio of two infinite sums involving the Fibonacci numbers. One such identity with a(n) is given in the formula section. Another is given in A394992 where further comments, links and references can be found.

Links

Hans J. H. Tuenter, Table of n, a(n) for n = 0..100

Formula

E.g.f.: (2/sqrt(5))*exp(-x/2)*sinh(sqrt(5)*x/2)/(1-x).

a(n) = n!*Sum_{k=0..n} F(-k)/k! = n!*Sum_{k=0..n} A039834(k)/k!.

a(n) = F(-n) + n*a(n-1) = (-1)^(n+1)*F(n) + n*a(n-1).

a(n+3) = (n+2)*a(n+2)+(n+3)*a(n+1)-(n+1)*a(n).

a(n) ~ sqrt(2*Pi/5)*n^(n+1/2)*exp(phi-1-n)*(1-exp(-sqrt(5))), where phi is A001622.

a(n)/n! ~ (exp(phi-1)-exp(-phi))/sqrt(5) = 2*sinh(sqrt(5)/2)/sqrt(5*e) = 0.741027921... = A099935 = A098689/e.

a(n) = Sum_{k=0..n} binomial(n,k)*(-1)^(n-k)*A111139(k).

A111139(n) = Sum_{k=0..n} binomial(n,k)*a(k).

A111139(n)/a(n) ~ e. For positive n, this ratio is strictly decreasing for n even and strictly increasing for n odd, both converging to the limit e.

Mathematica

a[n_] := n!*Sum[Fibonacci[-k]/k!, {k, 0, n}]; Table[a[n], {n, 0, 22}]

Crossrefs

Cf. A111139, A098689, A099935, A394992.

Cf. A000045, A039834, A001622 (phi), A001113 (e).

Sequence in context: A212291 A307176 A309175 * A179684 A250744 A197875

Adjacent sequences: A394990 A394991 A394992 * A394994 A394995 A394996

Keyword

nonn,easy

Author

Hans J. H. Tuenter, Apr 18 2026

Status

approved