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Expansion of e.g.f. (sinh(x) + 5*cosh(x) - 5)/(3 - 2*cosh(x)).
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%I #18 Apr 16 2019 02:51:37

%S 0,1,5,7,65,151,2105,6847,127265,532231,12365705,63206287,1762220465,

%T 10645162711,346257393305,2413453999327,89717615769665,

%U 708721089607591,29639206807284905,261679010699505967,12159552732032614865,118654880542567722871,6064946899313607640505

%N Expansion of e.g.f. (sinh(x) + 5*cosh(x) - 5)/(3 - 2*cosh(x)).

%F G.f.: Sum_{j>=1} j!*Lucas(j)*x^j / Product_{k=1..j} (1 + k*x).

%F a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n,k)*k!*Lucas(k).

%F a(n) ~ n! * (phi + (-1)^n/phi) / (2*log(phi))^(n+1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Apr 05 2019

%p a:=series((sinh(x)+5*cosh(x)-5)/(3-2*cosh(x)),x=0,23):seq(n!*coeff(a, x, n), n=0..22); # _Paolo P. Lava_, Apr 12 2019

%t nmax = 22; CoefficientList[Series[(Sinh[x] + 5 Cosh[x] - 5)/(3 - 2 Cosh[x]), {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 22; CoefficientList[Series[Sum[j! LucasL[j] x^j/Product[(1 + k x), {k, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]

%t Table[Sum[(-1)^(n - k) StirlingS2[n, k] k! LucasL[k], {k, 1, n}], {n, 0, 22}]

%o (PARI) my(x = 'x + O('x^30)); concat(0, Vec(serlaplace((sinh(x) + 5*cosh(x) - 5)/(3 - 2*cosh(x))))) \\ _Michel Marcus_, Apr 05 2019

%Y Cf. A000204, A050946, A105796, A105797, A263575.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 05 2019