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A323620 Expansion of e.g.f. 2*sqrt(1 + x)*sinh(sqrt(5)*log(1 + x)/2)/sqrt(5).

%I A323620 #14 Sep 08 2022 08:46:23
%S A323620 0,1,0,1,-4,19,-108,719,-5496,47465,-457160,4858865,-56490060,
%T A323620 713165035,-9715762980,142069257055,-2219386098160,36889108220305,
%U A323620 -650018185589520,12103669982341025,-237476572759473300,4896758300881695875,-105866710959427454300,2394660132226522508975
%N A323620 Expansion of e.g.f. 2*sqrt(1 + x)*sinh(sqrt(5)*log(1 + x)/2)/sqrt(5).
%H A323620 G. C. Greubel, <a href="/A323620/b323620.txt">Table of n, a(n) for n = 0..449</a>
%F A323620 a(n) = Sum_{k=0..n} Stirling1(n,k)*A000045(k).
%F A323620 From _Vaclav Kotesovec_, Jan 21 2019: (Start)
%F A323620 a(n) = -(-1)^n * cos(sqrt(5)*Pi/2) * (Gamma((3 + sqrt(5))/2) * Gamma(n - (1 + sqrt(5))/2) - Gamma((3 - sqrt(5))/2) * Gamma(n + (sqrt(5) - 1)/2)) / (Pi*sqrt(5)).
%F A323620 a(n) ~ -(-1)^n * n! / (sqrt(5) * Gamma((sqrt(5)-1)/2) * n^((3 - sqrt(5))/2)).
%F A323620 a(n) = -2*(n-2)*a(n-1) - (n^2 - 5*n + 5)*a(n-2). (End)
%t A323620 FullSimplify[nmax = 23; CoefficientList[Series[2 Sqrt[1 + x] Sinh[Sqrt[5] Log[1 + x]/2]/Sqrt[5], {x, 0, nmax}], x] Range[0, nmax]!]
%t A323620 Table[Sum[StirlingS1[n, k] Fibonacci[k], {k, 0, n}], {n, 0, 23}]
%o A323620 (PARI) {a(n) = sum(k=0,n, stirling(n,k,1)*fibonacci(k))};
%o A323620 vector(25, n, n--; a(n)) \\ _G. C. Greubel_, Feb 07 2019
%o A323620 (Magma) [(&+[StirlingFirst(n,k)*Fibonacci(k): k in [0..n]]): n in [0..25]]; // _G. C. Greubel_, Feb 07 2019
%o A323620 (Sage) [sum((-1)^(k+n)*stirling_number1(n,k)*fibonacci(k) for k in (0..n)) for n in (0..25)] # _G. C. Greubel_, Feb 07 2019
%Y A323620 Cf. A000045, A048994, A178840, A213593, A263576, A265165.
%K A323620 sign
%O A323620 0,5
%A A323620 _Ilya Gutkovskiy_, Jan 20 2019