A166356 Expansion of e.g.f. 1 + x*arctanh(x), even powers only.

1, 2, 8, 144, 5760, 403200, 43545600, 6706022400, 1394852659200, 376610217984000, 128047474114560000, 53523844179886080000, 26976017466662584320000, 16131658445064225423360000, 11292160911544957796352000000, 9146650338351415815045120000000

Offset: 0

Author

Paul Barry, Oct 12 2009

Keywords

• easy
• nonn

Comments

For n>0, (4*n-1)*a(n) corresponds to the number of random walk labelings of the friendship graph F_n (i.e., the one-point union of n triangles). - Sela Fried, May 20 2023

Links

• Sela Fried and Toufik Mansour, Further results on random walk labelings, arXiv:2305.09971 [math.CO], 2023.

Crossrefs

Cf. A053556, A001048, A068996, A143623.

Programs

Mathematica

a[0] = 1; a[n_] := (2*n - 1)! + (2*n - 2)!; Array[a, 14, 0] (* Amiram Eldar, Jan 02 2022 *)
With[{nn=40},Take[CoefficientList[Series[1+x ArcTanh[x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Apr 15 2024 *)

Formula

E.g.f.: 1 + x*arctanh(x) has expansion 1, 0, 2, 0, 8, 0, 144, ...

a(n) = (2n-1)! + (2n-2)! for n > 0; a(0) = 1.

a(n) -2*n*(2*n-3)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012

G.f.: 1 + x*G(0), where G(k) = 1 + 1/(1 - (k+2)*x/( (k+2)*x + (k+1)/((2*k+1)*(2*k+2))/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013

From Amiram Eldar, Jan 02 2022: (Start)

Sum_{n>=0} 1/a(n) = 2 - 1/e = 1 + A068996.

Sum_{n>=0} (-1)^n/a(n) = 2 - cos(1) - sin(1) = 2 - A143623. (End)