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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.
• Prabha Sivaraman Nair, A note on alternating weighted sums of Fibonacci numbers, Math. Montisnigri (2024) Vol. LX, 32-49. See p. 38.
• Prabha Sivaraman Nair and Rejikumar Karunakaran, On k-Fibonacci Brousseau Sums, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 8.
• R. L. Ollerton and A. G. Shannon, A Note on Brousseau's Summation Problem, Fibonacci Quart. 58 (2020), no. 5, 190-199.
• Daniele Parisse, On hypersequences of an arbitrary sequence and their weighted sums, Integers (2024) Vol. 24, Art. No. A70. See p. 25.
• Eric Weisstein's MathWorld, Polylogarithm.

Crossrefs

Cf. A000045, A000556, A005923, A136630, A320352, A358031, A358032.

Cf. A006154, A107403.

Programs

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 *)

PARI

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

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

a(n) = Sum_{k=0..n} 2^k * k! * A136630(n,k). - Seiichi Manyama, Jun 25 2025

Extensions

More terms from David W. Wilson