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Expansion of e.g.f. log(2*exp(x/2)*cosh(sqrt(5)*x/2) - 1).
0

%I #9 Jan 29 2019 05:01:15

%S 0,1,2,-3,-6,50,-13,-1498,6234,59145,-748678,-1415238,92962179,

%T -411570250,-11993577118,167710062977,1224967301754,-51920085859710,

%U 135335259830867,14992073315394822,-201575378391009366,-3667884891055854535,128570113943360964602,209758874692705861322

%N Expansion of e.g.f. log(2*exp(x/2)*cosh(sqrt(5)*x/2) - 1).

%F E.g.f.: log(1 + Sum_{k>=1} Lucas(k)*x^k/k!).

%F a(0) = 0; a(n) = Lucas(n) - (1/n)*Sum_{k=1..n-1} binomial(n,k)*Lucas(n-k)*k*a(k).

%p seq(n!*coeff(series(log(2*exp(x/2)*cosh(sqrt(5)*x/2)-1),x=0,24),x,n),n=0..23); # _Paolo P. Lava_, Jan 28 2019

%t nmax = 23; CoefficientList[Series[Log[2 Exp[x/2] Cosh[Sqrt[5] x/2] - 1], {x, 0, nmax}], x] Range[0, nmax]!

%t a[n_] := a[n] = LucasL[n] - Sum[Binomial[n, k] LucasL[n - k] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]

%Y Cf. A000032, A000204, A007553, A112005, A294222.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Jan 25 2019