A245242 a(n) = Sum_{k=0..n} binomial(n^2 - k^2, n*k - k^2).
1, 2, 5, 40, 987, 73026, 15656191, 9146092572, 15579632823935, 71399036100619112, 916371430754269894286, 33098484899485154272997507, 3182514246669584511131232330210, 875352526298195795986890973534420721, 650999500319874632196352991280266092913655
Offset: 0
Keywords
Examples
We can illustrate the terms as the row sums of triangle A245243; triangle A245243(n,k) = C(n^2 - k^2, n*k - k^2) begins: 1; 1, 1; 1, 3, 1; 1, 28, 10, 1; 1, 455, 495, 35, 1; 1, 10626, 54264, 8008, 126, 1; 1, 324632, 10518300, 4686825, 125970, 462, 1; 1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1; ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..67
Programs
-
Mathematica
Table[Sum[Binomial[n^2-k^2,n*k-k^2],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *) -
PARI
{a(n)=sum(k=0,n,binomial(n^2 - k^2, n*k - k^2))} for(n=0,20,print1(a(n),", ")) -
PARI
{a(n)=sum(k=0,n,binomial(n^2,n*k)*binomial(n*k,k^2)/binomial(n^2,k^2))} for(n=0,20,print1(a(n),", "))
Formula
a(n) = Sum_{k=0..n} C(n^2, n*k) * C(n*k, k^2) / C(n^2, k^2).
a(n) = Sum_{k=0..n} ((n+k)*(n-k))! / ( (n*(n-k))! * (k*(n-k))! ).
a(n) = Sum_{k=0..n} (n^2 - k^2)! / ( (n^2 - n*k)! * (n*k - k^2)! ).
Limit n->infinity a(n)^(1/n^2) = r^(-(1-r)^2/(2*r)) = 1.65459846190854391888257390278..., where r = 0.37667447497728449846981481128313080857... (see A245259) is the root of the equation r^(2*r-1) = (r+1)^(2*r). - Vaclav Kotesovec, Jul 15 2014
Comments