A000556 Expansion of exp(-x) / (1 - exp(x) + exp(-x)). (Formerly M3966 N1638)

1, 1, 5, 31, 257, 2671, 33305, 484471, 8054177, 150635551, 3130337705, 71556251911, 1784401334897, 48205833997231, 1402462784186105, 43716593539939351, 1453550100421124417, 51350258701767067711, 1920785418183176050505, 75839622064482770570791

Offset

0,3

References

Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin's summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..401

Gregory Dresden, On the Brousseau sums Sum_{i=1..n} i^p*Fibonacci(i), arxiv.org:2206.00115 [math.NT], 2022.

Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, Sums related to the Fibonacci sequence, Husson University (2021).

G. Ledin, Jr., On a certain kind of Fibonacci sums, Fib. Quart., 5 (1967), 45-58.

R. L. Ollerton and A. G. Shannon, A Note on Brousseau's Summation Problem, Fibonacci Quart. 58 (2020), no. 5, 190-199.

Eric Weisstein's MathWorld, Polylogarithm.

Eric Weisstein's MathWorld, Golden Ratio.

Eric Weisstein's MathWorld, Lucas Number.

Formula

a(n) = Sum_{k=0..n} k!*Fibonacci(k+1)*Stirling2(n,k).

E.g.f.: 1/(1 + U(0)) where U(k) = 1 - 2^k/(1 - x/(x - (k+1)*2^k/U(k+1) )); (continued fraction 3rd kind, 3-step ). - Sergei N. Gladkovskii, Dec 05 2012

a(n) ~ 2*n! / ((5+sqrt(5)) * log((1+sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, May 04 2015

a(n) = (-1)^(n+1)*(Li_{-n}(1-phi)*phi + Li_{-n}(phi)/phi)/sqrt(5), where Li_n(x) is the polylogarithm, phi=(1+sqrt(5))/2 is the golden ratio. - Vladimir Reshetnikov, Oct 30 2015

John W. Layman observes that this is also Sum (-2)^k*binomial(n, k)*b(n-k), where b() = A005923.

From Greg Dresden, May 13 2022 (Start):

For n > 0, a(n) = 1 + 2*Sum_{k=0..floor(n/2-1)} binomial(n,2*k+1) * a(n-2*k-1).

For n > 0, a(n) = Sum_{k=0..n-1} binomial(n,k)*A000557(k).

(End)

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*(2^j-1), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 05 2019

Mathematica

CoefficientList[Series[E^(-x)/(1-E^x+E^(-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, May 04 2015 *)
Round@Table[(-1)^(n+1) (PolyLog[-n, 1-GoldenRatio] GoldenRatio + PolyLog[-n, GoldenRatio]/GoldenRatio)/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 30 2015 *)

Prog

(PARI) a(n) = sum(k=0, n, k!*fibonacci(k+1)*stirling(n, k, 2)); \\ Michel Marcus, Oct 30 2015

Crossrefs

Cf. A005923, A216794.

Sequence in context: A126121 A167137 A279434 * A320512 A125598 A267436

Adjacent sequences: A000553 A000554 A000555 * A000557 A000558 A000559

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved