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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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 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 Cf. A007317, A000699, A088716. Sequence in context: A150092 A150093 A150094 * A234010 A360212 A121655 Adjacent sequences: A305115 A305116 A305117 * A305119 A305120 A305121 KEYWORD nonn AUTHOR Joerg Arndt, May 26 2018 STATUS approved

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Last modified August 6 07:30 EDT 2024. Contains 374960 sequences. (Running on oeis4.)