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A100872 a(n) = (1/sqrt(5)) * Sum_{k>0} k^(2n)/phi^(2k) where phi = (1+sqrt(5))/2 = A001622.

%I A100872 #33 Mar 15 2023 11:48:59
%S A100872 1,13,421,25453,2473141,352444093,69251478661,17943523153933,
%T A100872 5927841361456981,2431910546406522973,1212989379862721528101,
%U A100872 722875495525684291639213,507275965883448333971692021,414031618935013558427928710653,388884101194230308462039862028741
%N A100872 a(n) = (1/sqrt(5)) * Sum_{k>0} k^(2n)/phi^(2k) where phi = (1+sqrt(5))/2 = A001622.
%C A100872 A bisection of "Stirling-Bernoulli transform" of Fibonacci numbers.
%H A100872 Alois P. Heinz, <a href="/A100872/b100872.txt">Table of n, a(n) for n = 1..223</a>
%F A100872 a(n) = A050946(2*n).
%F A100872 From _Peter Bala_, Aug 20 2014: (Start)
%F A100872 E.g.f.: -1/2 + (1/2)*exp(z)/(3*exp(z) - exp(2*z) - 1) = z^2/2! + 13*z^4/4! + 421*z^6/6! + ....
%F A100872 a(n) = Sum_{k = 1..n} 2^(k-1)*A241171(n,k).
%F A100872 a(n) = Sum_{1 <= j <= k <= n} (-1)^(k-j)*binomial(2*k,k+j)*j^(2*n).
%F A100872 (End)
%t A100872 FullSimplify[Table[PolyLog[-2k, GoldenRatio^(-2)]/Sqrt[5], {k, 1, 10}]] (* _Vladimir Reshetnikov_, Feb 16 2011 *)
%t A100872 T[n_, k_] /; 1 <= k <= n := T[n, k] = k(2k-1) T[n-1, k-1] + k^2 T[n-1, k]; T[_, 1] = 1; T[_, _] = 0; a[n_] := Sum[2^(k-1) T[n, k], {k, 1, n}]; Array[a, 15] (* _Jean-François Alcover_, Jul 03 2019 *)
%o A100872 (PARI) a(n)=round(1/sqrt(5)*sum(k=1,500,k^(2*n)/((1+sqrt(5))/2)^(2*k)))
%Y A100872 Cf. A001622, A050946, A100868, A241171.
%K A100872 nonn,easy
%O A100872 1,2
%A A100872 _Benoit Cloitre_, Jan 08 2005