A209331 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, n*k-k^2).

1, 1, 2, 7, 86, 1905, 66002, 5218373, 1340847046, 688750226335, 527838995308056, 707409447204872377, 2844096719471817175298, 30274246332924074325724393, 517646331335208169889265781259, 13363896516779950029547538703868509

Offset: 0

Author

Paul D. Hanna, Mar 06 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..79

Crossrefs

Cf. A209330, A167009, A238696, A209428.

Programs

Mathematica

Table[Sum[Binomial[(n-k)^2, n*k-k^2], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)

PARI

{a(n)=sum(k=0,n\2, binomial((n-k)^2, n*k-k^2))}
for(n=0,20,print1(a(n),", "))

Formula

Equals the antidiagonal sums of triangle A209330(n,k) = C(n^2,n*k).

Limit n->infinity a(n)^(1/n^2) = ((1-r)/r)^((1-r)^2/(3-4*r)) = 1.4360944969025357119535113523184471047971386419..., where r = A323777 = 0.220676041323740696312822269998050167187681031... is the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r). - Vaclav Kotesovec, Mar 03 2014

Extensions

Name corrected by Vaclav Kotesovec, Mar 03 2014