A202139 Expansion of e.g.f. log(1/(1-artanh(x))).
0, 1, 1, 4, 14, 88, 544, 4688, 41712, 459520, 5333376, 71876352, 1027670016, 16428530688, 278818065408, 5167215464448, 101437811718144, 2140879726411776, 47698275298050048, 1130276555155243008, 28167446673847812096, 740796870212763254784
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..431
- Wikipedia contributors, Area function (inverse hyperbolic function), Wikipedia, the free encyclopedia, as of April 7, 2025.
Programs
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Mathematica
With[{nn=30},CoefficientList[Series[Log[1/(1-ArcTanh[x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 10 2022 *) -
Maxima
a(n):=n!*sum(((m-1)!*sum((stirling1(k+m,m)*2^k*binomial(n-1,k+m-1))/(k+m)!,k,0,n-m)),m,1,n);
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PARI
a_vector(n) = my(v=vector(n+1)); v[1]=0; for(i=1, n, v[i+1]=(i%2)*(i-1)!+sum(j=1, i\2, (2*j-2)!*binomial(i-1, 2*j-1)*v[i-2*j+2])); v; \\ Seiichi Manyama, Apr 30 2022
Formula
a(n) = n! * Sum_{m=1..n} (m-1)! * Sum_{k=0..n-m} Stirling1(k+m,m) * 2^k * binomial(n-1,k+m-1)/(k+m)!.
E.g.f.: log(2) - log(2 + log((1-x)/(1+x))). - Arkadiusz Wesolowski, Feb 19 2013
a(n) ~ n! * ((exp(2)+1)/(exp(2)-1))^n/n. - Vaclav Kotesovec, Jun 13 2013
a(0) = 0; a(n) = (n mod 2) * (n-1)! + Sum_{k=1..floor(n/2)} (2*k-2)! * binomial(n-1,2*k-1) * a(n-2*k+1). - Seiichi Manyama, Apr 30 2022
Extensions
Zero prepended by Harvey P. Dale, Sep 10 2022