A394993 - OEIS

%I #9 Apr 23 2026 00:18:17

%S 0,1,1,5,17,90,532,3737,29875,268909,2689035,29579474,354953544,

%T 4614396305,64601547893,969023219005,15504371503093,263574315554178,

%U 4744337679972620,90142415919483961,1802848318389672455,37859814686183132501,832915923096028897311

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

%C 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.

%H Hans J. H. Tuenter, <a href="/A394993/b394993.txt">Table of n, a(n) for n = 0..100</a>

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

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

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

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

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

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

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

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

%F 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.

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

%Y Cf. A111139, A098689, A099935, A394992.

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

%K nonn,easy

%O 0,4

%A _Hans J. H. Tuenter_, Apr 18 2026