A227403 a(n) = Sum_{k=0..n} binomial(n^2, n*k) * binomial(n*k, k^2).

1, 2, 14, 1514, 1308582, 17304263902, 1362702892177706, 1323407909279927430346, 11218363871234340925730020646, 637467717878006909442727527733810142, 519660435252919757259949810325837093364580014, 2289503386759572781844843312201361014103189493095636611

Offset: 0

Author

Paul D. Hanna, Sep 20 2013

Keywords

• nonn

Examples

			The following triangles illustrate the terms involved in the sum
a(n) = Sum_{k=0..n} A209330(n,k) * A228832(n,k).
Triangle A209330(n,k) = binomial(n^2, n*k) begins:
1;
1, 1;
1, 6, 1;
1, 84, 84, 1;
1, 1820, 12870, 1820, 1;
1, 53130, 3268760, 3268760, 53130, 1;
1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1;
...
Triangle A228832(n,k) = binomial(n*k, k^2) begins:
1;
1, 1;
1, 2, 1;
1, 3, 15, 1;
1, 4, 70, 220, 1;
1, 5, 210, 5005, 4845, 1;
1, 6, 495, 48620, 735471, 142506, 1; ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..46
• V. Kotesovec, Asymptotic of sequence A227403, Sep 21 2013

Crossrefs

Cf. A209330, A228832, A206849, A206152, A237421.

Programs

Mathematica

Table[Sum[Binomial[n^2,n*k]*Binomial[n*k,k^2],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Sep 21 2013 *)
r^(-(1+r)^2/(2*r))/.FindRoot[(1-r)^(2*r) == r^(2*r+1), {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n^2), Vaclav Kotesovec, Sep 21 2013 *)

PARI

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

Formula

a(n) = Sum_{k=0..n} (n^2)! / ( (n^2-n*k)! * (n*k-k^2)! * (k^2)! ).

Limit n->infinity a(n)^(1/n^2) = r^(-(1+r)^2/(2*r)) = 2.93544172048274..., where r = 0.6032326837741362... (see A237421) is the root of the equation (1-r)^(2*r) = r^(2*r+1). - Vaclav Kotesovec, Sep 21 2013