A298851 - OEIS

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

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OFFSET

0,3

LINKS

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

FORMULA

a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.774513671664430848697327843228386312953174421074432567764556466987... and c = 0.617929515483613293691991371141292259390065108300160936187723552669... - Vaclav Kotesovec, Feb 02 2018

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

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 *)

PROG

(Maxima) a(n):=if n<1 then 1 else 2*sum((n-k)^(4*n)/((2*n-k)!*k!*(-1)^k), k, 0, n);