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%I M1881 N0743 #80 Oct 26 2022 13:40:52
%S 1,2,8,50,416,4322,53888,783890,13031936,243733442,5064992768,
%T 115780447730,2887222009856,77998677862562,2269232452763648,
%U 70734934220015570,2351893466832306176,83086463910558199682,3107896091715557654528,122711086194279627711410
%N Expansion of e.g.f.: 1/(1-2*sinh(x)).
%C Inverse binomial transform of A005923. - _Vladimir Reshetnikov_, Oct 29 2015
%D Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin's summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A000557/b000557.txt">Table of n, a(n) for n = 0..200</a>
%H Gregory Dresden, <a href="https://arxiv.org/abs/2206.00115">On the Brousseau sums Sum_{i=1..n} i^p*Fibonacci(i)</a>, arxiv.org:2206.00115 [math.NT], 2022.
%H Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, <a href="https://www.researchgate.net/publication/350886459_SUMS_RELATED_TO_THE_FIBONACCI_SEQUENCE">Sums related to the Fibonacci sequence</a>, Husson University (2021).
%H G. Ledin, Jr., <a href="http://www.fq.math.ca/Scanned/5-1/ledin.pdf">On a certain kind of Fibonacci sums</a>, Fib. Quart., 5 (1967), 45-58.
%H R. L. Ollerton and A. G. Shannon, <a href="https://www.fq.math.ca/Papers1/58-5/ollerton.pdf">A Note on Brousseau's Summation Problem</a>, Fibonacci Quart. 58 (2020), no. 5, 190-199.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%F E.g.f.: 1/(1-2*sinh(x)). - _Vladeta Jovovic_, Jul 06 2002
%F a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(k,j)*(k-2*j)^n. - _Peter Luschny_, Jul 31 2011
%F a(n) = Sum_{k=0..n} k!*Stirling2(n, k)*Fibonacci(k+2).
%F a(n) ~ n! / (sqrt(5) * log((1+sqrt(5))/2)^(n+1)). - _Vaclav Kotesovec_, May 04 2015
%F 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
%F 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
%F Sum_{k=0..n-1} binomial(n,k)*a(k) = A000556(n). - _Greg Dresden_, Jun 01 2022
%F a(n) = A000556(n) + A320352(n). - _Seiichi Manyama_, Oct 26 2022
%p 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
%t f[n_] := Sum[ k!*StirlingS2[n, k]*Fibonacci[k + 2], {k, 0, n}]; Array[f, 20, 0] (* _Robert G. Wilson v_, Aug 16 2011 *)
%t With[{nn=20},CoefficientList[Series[1/(1-2*Sinh[x]),{x,0,nn}],x]Range[ 0,nn]!] (* _Harvey P. Dale_, Mar 11 2012 *)
%t Round@Table[(-1)^n (PolyLog[-n, 1-GoldenRatio]-PolyLog[-n, GoldenRatio])/Sqrt[5], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 29 2015 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(1/(1-2*sinh(x)))) \\ _Michel Marcus_, May 18 2022
%Y Cf. A000045, A000556, A005923, A320352, A358031, A358032.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_
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