A387042 a(n) = Sum_{k=0..n} binomial((n+k)*n, (n+k)*k).

1, 2, 22, 3500, 3903702, 66058817962, 6142112890141552, 6987021152124650220180, 67850336375886802959686935422, 4305222085343086827929862341832971717, 3923633842565310501037746535498640867868923722, 18947862261624122379177721509485181878924220314601284440

Offset

0,2

Links

Table of n, a(n) for n=0..11.

Vaclav Kotesovec, Graph - the asymptotic ratio (500 terms)

Formula

a(n) = Sum_{k=0..n} binomial((n+k)*n, (n+k)*k).

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

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

a(n) = Sum_{k=0..n} (2*n^2 - n*k)! / ( (2*n^2 - 3*n*k + k^2)! * (2*n*k - k^2)! ).

Limit_{n->oo} a(n)^(1/n^2) = r^(-(1+r)^2/(2*r)) = 2.93544172048274005711865243081001062972075994951247125..., where r = A237421. - Vaclav Kotesovec, Sep 24 2025

Example

The terms a(n) = Sum_{k=0..n} binomial((n+k)*n, (n+k)*k) begin
a(0) = 1;
a(1) = 1 + 1;
a(2) = 1 + 20 + 1;
a(3) = 1 + 495 + 3003 + 1;
a(4) = 1 + 15504 + 2704156 + 1184040 + 1;
a(5) = 1 + 593775 + 2319959400 + 62852101650 + 886163135 + 1; ...

Mathematica

Table[Sum[Binomial[(n + k)*n, (n + k)*k], {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Sep 24 2025 *)

Prog

(PARI) {a(n) = sum(k=0, n, binomial((n+k)*n, (n+k)*k) )}
for(n=0, 12, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, binomial((2*n-k)*n, (2*n-k)*k) )}
(PARI) {a(n) = sum(k=0, n, (n^2 + n*k)! / ( (n^2 - k^2)! * (n*k + k^2)! ) )}
(PARI) {a(n) = sum(k=0, n, (2*n^2 - n*k)! / ( (2*n^2 - 3*n*k + k^2)! * (2*n*k - k^2)! ) )}

Crossrefs

Cf. A167009, A245242, A228808, A237421.

Sequence in context: A261400 A113930 A181235 * A246852 A060601 A053952

Adjacent sequences: A387039 A387040 A387041 * A387043 A387044 A387045

Keyword

nonn

Author

Paul D. Hanna, Sep 19 2025

Status

approved