A298851 a(n) = [x^n] Product_{k=1..n} 1/(1-k^2*x).

1, 1, 21, 1408, 196053, 46587905, 16875270660, 8657594647800, 5974284925007685, 5336898188553325075, 5992171630749371157181, 8260051854943114812198756, 13714895317396748230146099660, 26998129079190909699998105620908, 62173633286588800021263427046090792

Offset: 0

Author

Seiichi Manyama, Feb 01 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A001044, A036969, A269945.

Cf. A007820, A299035, A299036.

Cf. A383853.

Programs

Maple

b:= proc(k, n) option remember; `if`(k=0, 1,
      add(b(k-1, j)*j^2, j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 19 2022

Mathematica

Table[SeriesCoefficient[Product[1/(1 - k^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 02 2018 *)
Join[{1}, Table[2*Sum[(-1)^(n-k) * Binomial[2*n, n-k] * k^(4*n), {k, 0, n}]/(2*n)!, {n, 1, 20}]] (* Vaclav Kotesovec, May 15 2025 *)

Maxima

a(n):=if n<1 then 1 else 2*sum((n-k)^(4*n)/((2*n-k)!*k!*(-1)^k),k,0,n);
makelist(a(n), n, 0, 20); /* Tani Akinari, Mar 09 2021 */

Formula

From Vaclav Kotesovec, Feb 02 2018, updated May 12 2025: (Start)

a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.774513671664430848697327843228386312953174421074432567764556466987... and c = 0.617929515483613293691991371141292259390065108300160936187723552669...

In closed form, a(n) ~ n^(2*n - 1/2) * r^(4*n + 1) / (sqrt(Pi*(2 - r^2)) * (r^2 - 1)^n * exp(2*n)), where r = 1.04438203376083348498401390634474776086902815721... is the root of the equation (1-r)/(1+r) = -exp(-4/r). (End)

a(n) = 2*(Sum_{k=0..n} (n-k)^(4*n)/((2*n-k)!*k!*(-1)^k)) for n>0. - Tani Akinari, Mar 09 2021

a(n) = A036969(2n,n) = A269945(2n,n). - Alois P. Heinz, Feb 19 2022

From Seiichi Manyama, May 12 2025: (Start)

a(n) = Sum_{k=0..2*n} (-n)^k * binomial(4*n,k) * Stirling2(4*n-k,2*n).

a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(k+n,n) * Stirling2(3*n-k,n). (End)