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.
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.
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
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved