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

%I A166356 #28 Apr 15 2024 20:20:35
%S A166356 1,2,8,144,5760,403200,43545600,6706022400,1394852659200,
%T A166356 376610217984000,128047474114560000,53523844179886080000,
%U A166356 26976017466662584320000,16131658445064225423360000,11292160911544957796352000000,9146650338351415815045120000000
%N A166356 Expansion of e.g.f. 1 + x*arctanh(x), even powers only.
%C A166356 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
%H A166356 Sela Fried and Toufik Mansour, <a href="https://arxiv.org/abs/2305.09971">Further results on random walk labelings</a>, arXiv:2305.09971 [math.CO], 2023.
%F A166356 E.g.f.: 1 + x*arctanh(x) has expansion 1, 0, 2, 0, 8, 0, 144, ...
%F A166356 a(n) = (2n-1)! + (2n-2)! for n > 0; a(0) = 1.
%F A166356 a(n) -2*n*(2*n-3)*a(n-1) = 0. - _R. J. Mathar_, Nov 24 2012
%F A166356 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
%F A166356 From _Amiram Eldar_, Jan 02 2022: (Start)
%F A166356 Sum_{n>=0} 1/a(n) = 2 - 1/e = 1 + A068996.
%F A166356 Sum_{n>=0} (-1)^n/a(n) = 2 - cos(1) - sin(1) = 2 - A143623. (End)
%t A166356 a[0] = 1; a[n_] := (2*n - 1)! + (2*n - 2)!; Array[a, 14, 0] (* _Amiram Eldar_, Jan 02 2022 *)
%t A166356 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 *)
%Y A166356 Cf. A053556, A001048, A068996, A143623.
%K A166356 easy,nonn
%O A166356 0,2
%A A166356 _Paul Barry_, Oct 12 2009