A129763 a(n) = Sum_{k=1..n} binomial(n+k-1, n)^2 / n.

1, 5, 39, 369, 3898, 44239, 528083, 6544745, 83496720, 1090091650, 14501708246, 195954553755, 2682953977174, 37150480629539, 519455719162283, 7325383709872345, 104080732316126716, 1488685017986884528, 21420051312840487968

Offset: 1

Author

Alexander Adamchuk, May 15 2007

Keywords

• nonn

Comments

Note that Sum_{k=1..n} binomial(n+k-1, n) / n = Catalan(n) = A000108(n).

p divides a((p-1)/2) for prime p = {5, 13, 17, 29, 37, 41, 53, ...} = A002144 Pythagorean primes: primes of the form 4*k + 1. - Alexander Adamchuk, Dec 27 2013

Crossrefs

Cf. A000108 (Catalan numbers), A002144 (Pythagorean primes).

Programs

Maple

ogf := (4-x)^(1/2)*x^(-3/2)*Int((x+5/4)*hypergeom([1/2, 1/2],[1],16*x)/((4-x)^(3/2)*x^(1/2)),x) - 5/(8*x);
series(eval(ogf, Int = proc(a,x) int(series(a,x=0,30),x) end), x=0, 30); # Mark van Hoeij, May 05 2013

Mathematica

Table[ Sum[ Binomial[ n+k-1, n ]^2, {k,1,n} ] / n, {n,1,30} ]

Formula

G.f.: has an anti-derivative of a hypergeometric function, see Maple program. - Mark van Hoeij, May 05 2013

Recurrence: 2*n^2*(2*n + 1)*(21*n^2 - 62*n + 46)*a(n) = (1365*n^5 - 6067*n^4 + 9948*n^3 - 7478*n^2 + 2640*n - 360)*a(n-1) - 4*(n-2)*(2*n - 3)^2*(21*n^2 - 20*n + 5)*a(n-2). - Vaclav Kotesovec, Mar 02 2014

a(n) ~ 16^n / (3*Pi*n^2). - Vaclav Kotesovec, Mar 02 2014