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URL: https://oeis.org/A305118

⇱ A305118 - OEIS

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A305118
a(n) = Sum_{k=0..n-1} ( 1 + a(k) * a(n-k-1) ) for n >= 1, a(0) = 1.
1
1, 2, 6, 19, 66, 249, 996, 4148, 17784, 77939, 347516, 1571304, 7187288, 33196887, 154611392, 725284721, 3423760262, 16251813715, 77523741208, 371428985796, 1786623827240, 8624669381161, 41769772877288, 202893913979291, 988224403828490, 4825331506973445
OFFSET
0,2
LINKS
FORMULA
G.f. A(x) satisfies: A(x) = 1 + x * (1/(1 - x)^2 + A(x)^2). - Ilya Gutkovskiy, Jun 30 2020
From Vaclav Kotesovec, Jun 30 2020: (Start)
G.f.: (1 - sqrt(1 - 4*x*(1 + x/(1 - x)^2))) / (2*x).
a(n) ~ sqrt(1/r + 2/(1 - r)^3) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.19288682865259090392018... is the real root of the equation -1 + 6*r - 5*r^2 + 4*r^3 = 0. (End)
MATHEMATICA
CoefficientList[Series[(1 - Sqrt[1 - 4*x*(1 + x/(1 - x)^2)]) / (2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 30 2020 *)
PROG
(PARI)
seq(N) = {
my(a = vector(N)); a[1] = 1;
for (n=2, N, a[n] = sum(k=1, n-1, 1 + a[k]*a[n-k])); a;
};
seq(32)
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt, May 26 2018
STATUS
approved