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

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..67

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

Example

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; ...

Mathematica

Table[Sum[Binomial[n^2-k^2, n*k-k^2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 15 2014 *)

Prog

(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), ", "))

Crossrefs

Cf. A245243, A227403, A245259.

Sequence in context: A183910 A162035 A366358 * A004096 A156152 A106885

Adjacent sequences: A245239 A245240 A245241 * A245243 A245244 A245245

Keyword

nonn

Author

Paul D. Hanna, Jul 14 2014

Status

approved