A270709 a(n) = (n+1)*Sum_{k=0..(n-1)/2} (binomial(k+1,n-2*k-1)*binomial(2*k,k)/(k+1)^2).

0, 2, 3, 2, 5, 7, 14, 26, 51, 103, 209, 435, 910, 1930, 4122, 8874, 19227, 41893, 91751, 201839, 445841, 988403, 2198547, 4905147, 10974210, 24615134, 55341636, 124694354, 281525678, 636802626, 1442953404, 3274997130, 7444505615, 16946749249

Offset: 0

Author

Vladimir Kruchinin, Mar 22 2016

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108.

Programs

Mathematica

Table[(n+1)*Sum[Binomial[k+1,n-2*k-1] * Binomial[2*k,k] / (k+1)^2, {k,0,(n-1)/2}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 22 2016 *)

Maxima

makelist(coeff(taylor(((3*x+2)*(1-sqrt(1-4*(x^3+x^2))))/(2*(x^2+x)),x,0,15),x,n),n,0,15);
a(n):=(n+1)*sum((binomial(k+1,n-2*k-1)*binomial(2*k,k))/(k+1)^2,k,0,(n-1)/2);

PARI

x='x+O('x^200); concat(0, Vec(((3*x+2)*(1-sqrt(1-4*(x^3+x^2))))/(2*(x^2+x)))) \\ Altug Alkan, Mar 22 2016

Formula

G.f.: ((3*x+2)*(1-sqrt(1-4*(x^3+x^2))))/(2*(x^2+x)).

a(n) ~ (3*r+2) * sqrt(3-4*r^2) * 2^(2*n+2) * r^(n+3) * (r+1)^(n+1) / (n^(3/2) * sqrt(Pi)), where r = 0.41964337760708... is the real root of the equation 4*r^2*(1+r) = 1. - Vaclav Kotesovec, Mar 22 2016