A307361 Expansion of e.g.f. (sinh(x) + 5*cosh(x) - 5)/(3 - 2*cosh(x)).
0, 1, 5, 7, 65, 151, 2105, 6847, 127265, 532231, 12365705, 63206287, 1762220465, 10645162711, 346257393305, 2413453999327, 89717615769665, 708721089607591, 29639206807284905, 261679010699505967, 12159552732032614865, 118654880542567722871, 6064946899313607640505
Offset: 0
Keywords
Programs
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Maple
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
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Mathematica
nmax = 22; CoefficientList[Series[(Sinh[x] + 5 Cosh[x] - 5)/(3 - 2 Cosh[x]), {x, 0, nmax}], x] Range[0, nmax]! nmax = 22; CoefficientList[Series[Sum[j! LucasL[j] x^j/Product[(1 + k x), {k, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x] Table[Sum[(-1)^(n - k) StirlingS2[n, k] k! LucasL[k], {k, 1, n}], {n, 0, 22}] -
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
Formula
G.f.: Sum_{j>=1} j!*Lucas(j)*x^j / Product_{k=1..j} (1 + k*x).
a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n,k)*k!*Lucas(k).
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