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A202139 Expansion of e.g.f. log(1/(1-artanh(x))).

%I A202139 #40 Apr 16 2025 10:23:42
%S A202139 0,1,1,4,14,88,544,4688,41712,459520,5333376,71876352,1027670016,
%T A202139 16428530688,278818065408,5167215464448,101437811718144,
%U A202139 2140879726411776,47698275298050048,1130276555155243008,28167446673847812096,740796870212763254784
%N A202139 Expansion of e.g.f. log(1/(1-artanh(x))).
%H A202139 Seiichi Manyama, <a href="/A202139/b202139.txt">Table of n, a(n) for n = 0..431</a>
%H A202139 Wikipedia contributors, <a href="https://en.wikipedia.org/wiki/Area_function_(inverse_hyperbolic_function)">Area function (inverse hyperbolic function)</a>, Wikipedia, the free encyclopedia, as of April 7, 2025.
%F A202139 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)!.
%F A202139 E.g.f.: log(2) - log(2 + log((1-x)/(1+x))). - _Arkadiusz Wesolowski_, Feb 19 2013
%F A202139 a(n) ~ n! * ((exp(2)+1)/(exp(2)-1))^n/n. - _Vaclav Kotesovec_, Jun 13 2013
%F A202139 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
%t A202139 With[{nn=30},CoefficientList[Series[Log[1/(1-ArcTanh[x])],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Sep 10 2022 *)
%o A202139 (Maxima)
%o A202139 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);
%o A202139 (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
%Y A202139 Cf. A003704, A226968.
%K A202139 nonn
%O A202139 0,4
%A A202139 _Vladimir Kruchinin_, Dec 12 2011
%E A202139 Zero prepended by _Harvey P. Dale_, Sep 10 2022