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%I M2953 #43 May 16 2022 16:21:52
%S 1,3,13,81,673,6993,87193,1268361,21086113,394368993,8195330473,
%T 187336699641,4671623344753,126204511859793,3671695236949753,
%U 114451527759954921,3805443567253430593,134436722612325267393,5028681509898733705033,198550708258762398282201
%N From solution to a difference equation.
%C Binomial transform of A000557. - _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. See p. 49.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H P. R. J. Asveld & N. J. A. Sloane, <a href="/A005442/a005442.pdf">Correspondence, 1987</a>
%H P. R. J. Asveld, <a href="http://www.fq.math.ca/Scanned/25-4/asveld.pdf">A family of Fibonacci-like sequences</a>, Fib. Quart., 25 (1987), 81-83.
%F E.g.f.: exp(x)/(1-2*sinh(x)). - Sander Zwegers (s.zwegers(AT)hetnet.nl), Jun 28 2007
%F E.g.f.: 1/( U(0) -1 ) where U(k) = 1 + 1/(2^k - 2*x*4^k/(2*x*2^k - (k+1)/U(k+1) )); (continued fraction 3rd kind, 3-step ). - _Sergei N. Gladkovskii_, Dec 05 2012
%F a(n) ~ n! * phi / (sqrt(5) * (log(phi))^(n+1)), where phi is the golden ratio. - _Vaclav Kotesovec_, Nov 27 2017
%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k + 1) * binomial(n,k) * (2^k + 1) * a(n-k). - _Ilya Gutkovskiy_, Jan 16 2020
%F a(n) = A000556(n) + A000557(n) for n>0. - _Greg Dresden_, May 13 2022
%t Round@Table[Sum[Binomial[n, k] (-1)^k (PolyLog[-k, 1-GoldenRatio] - PolyLog[-k, GoldenRatio])/Sqrt[5] , {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 29 2015 *)
%Y Cf. A000556, A000557.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Nov 23 2001
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