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A337555
a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n,k) * (3^k-1) * a(n-k).
2
1, 1, 6, 43, 408, 4861, 69516, 1159663, 22108848, 474192601, 11300589876, 296237533483, 8471642214888, 262456441714741, 8756520140416236, 313017838828154503, 11935355244756882528, 483537933291091103281, 20741938090482567562596, 939180816648348685174723
OFFSET
0,3
FORMULA
E.g.f.: 2 / (2 + exp(x) - exp(3*x)).
a(n) ~ n! / ((r+3) * log(r)^(n+1)), where r = 1.52137970680456756960408... is the real root of the equation r^3 - r = 2. - Vaclav Kotesovec, Aug 31 2020
MATHEMATICA
a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n, k] (3^k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[2/(2 + Exp[x] - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
PROG
(PARI) seq(n)={Vec(serlaplace(2 / (2 + exp(x + O(x*x^n)) - exp(3*x + O(x*x^n)))))} \\ Andrew Howroyd, Aug 31 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 31 2020
STATUS
approved