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A367946
Expansion of e.g.f. exp(2*(exp(2*x) - 1) + x).
1
1, 5, 33, 261, 2369, 24069, 269153, 3272453, 42858113, 600181765, 8933677729, 140645797125, 2332169258945, 40586333768197, 738998405168609, 14040304543111941, 277678389593341185, 5704502830382733317, 121500343635119818017, 2678407616841000605957, 61015572313688043492929
OFFSET
0,2
FORMULA
G.f. A(x) satisfies: A(x) = 1 + x * ( A(x) + 4 * A(x/(1 - 2*x)) / (1 - 2*x) ).
a(n) = exp(-2) * Sum_{k>=0} 2^k * (2*k+1)^n / k!.
a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 2^(k+1) * a(n-k).
MATHEMATICA
nmax = 20; CoefficientList[Series[Exp[2 (Exp[2 x] - 1) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(2*x) - 1) + x))) \\ Michel Marcus, Dec 07 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 05 2023
STATUS
approved