A049214 - OEIS

%I #22 Aug 13 2024 14:36:45

%S 1,20,784,52352,5360256,782525952,154594381824,39746508226560,

%T 12902483299368960,5162443736924160000,2496471943395999744000,

%U 1435556519572510605312000,968234590214616380866560000

%N Scaled coefficients of (arctanh x)^3.

%F E.g.f.: (arctanh x)^3 or (1-x^2)^-1 * (arctanh x)^2.

%F a(n) is coefficient of x^(2*n+3) in (arctanh x)^3, multiplied by (2*n+3)!/6.

%F a(n) = (1/4)* Sum_{p=1..2*n+2} binomial(2*n+1,p-1) Sum_{m=3..2*n+3} binomial(m,3)*abs(s(2*n+3,m))*((-p)^(m-3)+(-p)^(m-3)) where s(n,m) is Stirling number of first kind . - _Ammar Khatab_, Aug 13 2024

%e (arctanh x)^3 = x^3 + x^5 + 14/15*x^7 + 818/945*x^9 + ...

%t a[n_] := SeriesCoefficient[ArcTanh[x]^3, {x, 0, 2n+3}]*(2n+3)!/6;

%t a /@ Range[0, 12] (* _Jean-François Alcover_, Mar 26 2021 *)

%Y Cf. A049215, A049216, A049217.

%K nonn

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)