A337168 a(n) = (-1)^n + 2 * Sum_{k=0..n-1} a(k) * a(n-k-1).

1, 1, 5, 21, 105, 553, 3053, 17405, 101713, 606033, 3667797, 22485477, 139340985, 871429497, 5492959293, 34862161869, 222592918689, 1428814897825, 9215016141989, 59684122637237, 388045493943049, 2531696701375689, 16569559364596365, 108758426952823709

Offset: 0

Author

Ilya Gutkovskiy, Jan 28 2021

Keywords

• nonn

Comments

Inverse binomial transform of A151374.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See pp. 97-98.

Crossrefs

Cf. A000108, A005043, A052701, A151374, A162326, A337169.

Programs

Mathematica

a[n_] := a[n] = (-1)^n + 2 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k CatalanNumber[k], {k, 0, n}], {n, 0, 23}]
Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 8], {n, 0, 23}]

Formula

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

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

E.g.f.: exp(3*x) * (BesselI(0,4*x) - BesselI(1,4*x)).

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * Catalan(k).

a(n) ~ 7^(n + 3/2) / (2^(9/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021