A038602 One half of convolution of central binomial coefficients A000984(n) with A000984(n+2), n >= 0.

3, 16, 73, 316, 1334, 5552, 22901, 93892, 383290, 1559680, 6331098, 25649976, 103758828, 419195552, 1691825933, 6822051092, 27488564498, 110691186272, 445487285678, 1792047789512, 7205785665908, 28963557761312

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Also convolution of A000346 with Catalan numbers but with C(0)=1 replaced by 3

Links

• Fung Lam, Table of n, a(n) for n = 0..1500

Crossrefs

Cf. A000108, A000346.

Programs

Mathematica

CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x)*((1-Sqrt[1-4*x])/(2*x)+2)/(1-4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 28 2014 *)

Formula

a(n) = 2^(2*n+3)-(3*n+5)*C(n+1), C(n): Catalan numbers A000108.

G.f.: c(x)*(c(x)+2)/(1-4*x), where c(x) is G.f. for Catalan numbers.

a(n) ~ 2^(2*n+3) * (1-3/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Mar 28 2014

Recurrence: n*(n+2)*a(n) = 2*(4*n^2 + 5*n - 1)*a(n-1) - 8*(n+1)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Mar 28 2014