A229052 a(n) = Sum_{k=0..n} binomial(n^2-n*k, n*k-k^2) * binomial(n*k, k^2).

1, 2, 6, 92, 6662, 2150552, 3093730764, 18251332286098, 466740831542894470, 47238803741195397513182, 20522607409110459026633535856, 34700017072200465774261952422246668, 250699892545838622857396499800167790109260, 6984916990466628202550631436961441381064765905022

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..50

Formula

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

a(n) = Sum_{k=0..n} A228832(n, n-k) * A228832(n, k).

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

a(n) ~ c * 2^(n^2+2)/(Pi*n^2), where c = EllipticTheta[3,0,1/E^2] = 1.271341522189... if n is even and c = EllipticTheta[2,0,1/E^2] = 1.23528676585389... if n is odd. - Vaclav Kotesovec, Sep 22 2013

Example

The triangle A228832(n,k) = C(n*k, k^2) illustrates the terms involved in the sum a(n) = Sum_{k=0..n} A228832(n, n-k) * A228832(n, k):
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;
1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1; ...

Mathematica

Table[Sum[Binomial[n^2 - n k, n k - k^2] Binomial[n k, k^2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 22 2013 *)

Prog

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

Crossrefs

Cf. A228832, A206847, A218792.

Sequence in context: A343021 A007188 A206156 * A363838 A280117 A129364

Adjacent sequences: A229049 A229050 A229051 * A229053 A229054 A229055

Keyword

nonn

Author

Paul D. Hanna, Sep 22 2013

Status

approved