A000557 Expansion of e.g.f.: 1/(1-2*sinh(x)). (Formerly M1881 N0743)

1, 2, 8, 50, 416, 4322, 53888, 783890, 13031936, 243733442, 5064992768, 115780447730, 2887222009856, 77998677862562, 2269232452763648, 70734934220015570, 2351893466832306176, 83086463910558199682, 3107896091715557654528, 122711086194279627711410

Offset

0,2

Comments

Inverse binomial transform of A005923. - Vladimir Reshetnikov, Oct 29 2015

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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.

Formula

E.g.f.: 1/(1-2*sinh(x)). - Vladeta Jovovic, Jul 06 2002

a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(k,j)*(k-2*j)^n. - Peter Luschny, Jul 31 2011

a(n) = Sum_{k=0..n} k!*Stirling2(n, k)*Fibonacci(k+2).

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

a(n) = (-1)^n*(Li_{-n}(1-phi)-Li_{-n}(phi))/sqrt(5), where Li_n(x) denotes the polylogarithm, phi=(1+sqrt(5))/2. - Vladimir Reshetnikov, Oct 29 2015

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022

Sum_{k=0..n-1} binomial(n,k)*a(k) = A000556(n). - Greg Dresden, Jun 01 2022

a(n) = A000556(n) + A320352(n). - Seiichi Manyama, Oct 26 2022

Maple

A000557 := proc(n) local k, j; add(add((-1)^j*binomial(k, j)*(k-2*j)^n, j=0..k), k=0..n) end: # Peter Luschny, Jul 31 2011

Mathematica

f[n_] := Sum[ k!*StirlingS2[n, k]*Fibonacci[k + 2], {k, 0, n}]; Array[f, 20, 0] (* Robert G. Wilson v, Aug 16 2011 *)
With[{nn=20}, CoefficientList[Series[1/(1-2*Sinh[x]), {x, 0, nn}], x]Range[ 0, nn]!] (* Harvey P. Dale, Mar 11 2012 *)
Round@Table[(-1)^n (PolyLog[-n, 1-GoldenRatio]-PolyLog[-n, GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(1/(1-2*sinh(x)))) \\ Michel Marcus, May 18 2022

Crossrefs

Cf. A000045, A000556, A005923, A320352, A358031, A358032.

Sequence in context: A274273 A121677 A120956 * A193352 A002801 A322738

Adjacent sequences: A000554 A000555 A000556 * A000558 A000559 A000560

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson

Status

approved