%I #18 Jan 27 2019 06:17:14

%S 1,1,2,7,86,1905,66002,5218373,1340847046,688750226335,

%T 527838995308056,707409447204872377,2844096719471817175298,

%U 30274246332924074325724393,517646331335208169889265781259,13363896516779950029547538703868509

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

%H G. C. Greubel, <a href="/A209331/b209331.txt">Table of n, a(n) for n = 0..79</a>

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

%F 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

%t Table[Sum[Binomial[(n-k)^2, n*k-k^2], {k, 0, Floor[n/2]}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 03 2014 *)

%o (PARI) {a(n)=sum(k=0,n\2, binomial((n-k)^2, n*k-k^2))}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A209330, A167009, A238696, A209428.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 06 2012

%E Name corrected by _Vaclav Kotesovec_, Mar 03 2014