A354733 a(0) = a(1) = 1; a(n) = 2 * Sum_{k=0..n-2} a(k) * a(n-k-2).

1, 1, 2, 4, 10, 24, 64, 168, 464, 1280, 3624, 10304, 29728, 86240, 252480, 743040, 2200640, 6547200, 19571200, 58727680, 176883200, 534476800, 1619912320, 4923070464, 14999764480, 45807916544, 140196076544, 429931051008, 1320905583616, 4065358827520

Offset: 0

Author

Ilya Gutkovskiy, Jun 04 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A007477, A052701, A151374, A354734, A354735, A354736.

Programs

Mathematica

a[0] = a[1] = 1; a[n_] := a[n] = 2 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 29}]
nmax = 29; CoefficientList[Series[(1 - Sqrt[1 - 8 x^2 (1 + x)])/(4 x^2), {x, 0, nmax}], x]

PARI

a(n) = sum(k=0, n\2, 2^k*binomial(k+1, n-2*k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Nov 05 2023

Formula

G.f. A(x) satisfies: A(x) = 1 + x + 2 * (x * A(x))^2.

G.f.: (1 - sqrt(1 - 8 * x^2 * (1 + x))) / (4 * x^2).

a(n) ~ 5^(1/4) * (1 + sqrt(5))^(n+2) / (8 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 04 2022

a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(k+1,n-2*k) * A000108(k). - Seiichi Manyama, Nov 05 2023