A049216 Scaled coefficients of (arctanh x)^5.

1, 70, 6384, 804320, 136804096, 30459752960, 8632830664704, 3041109959196672, 1305140879116763136, 670935549630120394752, 407240889859179425562624, 288272814806050917816729600

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn

Examples

			(arctanh x)^5 = x^5 + 5/3*x^7 + 19/9*x^9 + 457/189*x^11 + ...
		

Crossrefs

Cf. A049214, A049215, A049217.

Programs

Mathematica

a[n_] := SeriesCoefficient[ArcTanh[x]^5, {x, 0, 2n+5}]*(2n+5)!/5!;
a /@ Range[0, 11] (* Jean-François Alcover, Mar 26 2021 *)

Formula

E.g.f.: (arctanh x)^5 or (1-x^2)^-1 * (arctanh x)^4.

a(n) is coefficient of x^(2*n+5) in (arctanh x)^5, multiplied by (2*n+5)!/5!.

a(n) = (1/16 ) * Sum_{p=1..2*n+4} binomial(2*n+3,p-1)*Sum_{m=5..2*n+5} binomial(m,5)*abs(s(2*n+5,m))*((-p)^(m-5)+(-p)^(m-5)) where s(n,m) is Stirling number of first kind . - Ammar Khatab, Aug 13 2024