A346543 a(n) = [x^n] Product_{k=1..2*n} (x + (2*k-1)^2).

1, 10, 1974, 1234948, 1601489318, 3541644282540, 11934462103156540, 56947950742822581960, 365458809637016986262790, 3035813466162156094097686300, 31694033885101849517370941522644, 406222401519003083851664224927890360, 6271146756206887832796744632163811733084

Offset: 0

Author

Seiichi Manyama, Sep 27 2021

Keywords

• nonn

Examples

			(1/3!) * (arcsin(x))^3 = x^3/3! + 10 * x^5/5! + ... . So a(1) =10.
(1/5!) * (arcsin(x))^5 = x^5/5! + 35 * x^7/7! + 1974 * x^9/9! + ... . So a(2) = 1974.
		

Crossrefs

Cf. A008956, A016754, A234324, A293318.

Programs

Mathematica

Table[SeriesCoefficient[Product[(x + (2*k-1)^2), {k, 1, 2*n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *)

PARI

a(n) = polcoef(prod(k=1, 2*n, x+(2*k-1)^2), n);

Formula

a(n) = A008956(2*n,n).

a(n) = (4*n+1)! * [x^(4*n+1)] (1/(2*n+1)!) * (arcsin(x))^(2*n+1).

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 121.8904568356133798202328777176879971969471503678428704459083316116687149... and c = 0.1081647814943965981694666415038643176470488612855594762896553127... - Vaclav Kotesovec, Oct 16 2021