A336873 a(n) = Sum_{k=0..n} (binomial(n+k,k) * binomial(n,k))^n.

1, 3, 73, 36729, 473940001, 155741521320033, 1453730786373283012225, 415588116056535702096428038017, 3278068950996636050857475073848209555969, 756475486389705843580676191270930552553654909184513, 5850304627708628483969594929628923064185219454493588333628772353

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..37

Vaclav Kotesovec, Plot of a(n)/(binomial(n + n/sqrt(2),n/sqrt(2)) * binomial(n,n/sqrt(2)))^n for n = 1..500

Formula

a(n)^(1/n) ~ (1 + sqrt(2))^(2*n + 1) / (Pi*sqrt(2)*n). - Vaclav Kotesovec, Jul 10 2021

Mathematica

a[n_] := Sum[(Binomial[n+k, k] * Binomial[n, k])^n, {k, 0, n} ]; Array[a, 11, 0] (* Amiram Eldar, Aug 06 2020 *)

Prog

(PARI) {a(n) = sum(k=0, n, (binomial(n+k, k)*binomial(n, k))^n)}
(Magma) [(&+[(Binomial(2*j, j)*Binomial(n+j, n-j))^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022
(SageMath)
def A336873(n): return sum((binomial(2*j, j)*binomial(n+j, n-j))^n for j in (0..n))
[A336873(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

Crossrefs

Cf. A001850, A005259, A092813, A092814, A092815, A167010, A218689, A336829, A346200.

Sequence in context: A307232 A002667 A145675 * A121981 A337413 A337409

Adjacent sequences: A336870 A336871 A336872 * A336874 A336875 A336876

Keyword

nonn

Author

Seiichi Manyama, Aug 06 2020

Status

approved