A337387 a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

1, 7, 74, 1175, 24310, 610897, 17920356, 598099077, 22305598630, 917158184525, 41148369048876, 1997720107411613, 104241356841544636, 5813083330109559415, 344783011379207286920, 21660231928192698604995, 1436143861200146476260102, 100179915387243084700279349

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

From Vaclav Kotesovec, Aug 31 2020: (Start)

a(n) ~ (2 + sqrt(n))^(2*n + 3/2) / (2*n*sqrt(2*Pi)).

a(n) ~ exp(4*sqrt(n) - 4) * n^(n - 1/4) / sqrt(8*Pi) * (1 + 25/(3*sqrt(n)) + 427/(18*n)). (End)

Mathematica

a[n_] := Sum[If[n == 0, Boole[n == k], n^(n - k)] * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Aug 25 2020 *)

Prog

(PARI) {a(n) = sum(k=0, n, n^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

Crossrefs

Main diagonal of A337369.

Cf. A337388.

Sequence in context: A266305 A098118 A097821 * A054745 A323322 A356589

Adjacent sequences: A337384 A337385 A337386 * A337388 A337389 A337390

Keyword

nonn

Author

Seiichi Manyama, Aug 25 2020

Status

approved