A049214 Scaled coefficients of (arctanh x)^3.

1, 20, 784, 52352, 5360256, 782525952, 154594381824, 39746508226560, 12902483299368960, 5162443736924160000, 2496471943395999744000, 1435556519572510605312000, 968234590214616380866560000

Offset

0,2

Links

Table of n, a(n) for n=0..12.

Formula

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

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

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

Example

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

Mathematica

a[n_] := SeriesCoefficient[ArcTanh[x]^3, {x, 0, 2n+3}]*(2n+3)!/6;
a /@ Range[0, 12] (* Jean-François Alcover, Mar 26 2021 *)

Crossrefs

Cf. A049215, A049216, A049217.

Sequence in context: A034404 A179712 A012802 * A002455 A322890 A267671

Adjacent sequences: A049211 A049212 A049213 * A049215 A049216 A049217

Keyword

nonn

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved