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A296676 Expansion of e.g.f. 1/(1 - arctanh(x)).

%I A296676 #19 Jun 30 2025 10:04:10
%S A296676 1,1,2,8,40,264,2048,18864,196992,2330112,30519552,440998656,
%T A296676 6940852224,118501542912,2177222879232,42886017982464,900748014944256,
%U A296676 20107190510714880,475167358873239552,11854636521914695680,311291779253770911744,8583598112533040332800,247944624171011289907200
%N A296676 Expansion of e.g.f. 1/(1 - arctanh(x)).
%H A296676 Alois P. Heinz, <a href="/A296676/b296676.txt">Table of n, a(n) for n = 0..430</a>
%F A296676 E.g.f.: 1/(1 + (log(1 - x) - log(1 + x))/2).
%F A296676 a(n) ~ n! * 4*exp(2) * (exp(2)+1)^(n-1) / (exp(2)-1)^(n+1). - _Vaclav Kotesovec_, Dec 18 2017
%F A296676 a(n) = Sum_{k=0..n} k! * A111594(n,k). - _Seiichi Manyama_, Jun 30 2025
%e A296676 1/(1 - arctanh(x)) = 1 + x/1! + 2*x^2/2! + 8*x^3/3! + 40*x^4/4! + 264*x^5/5! + ...
%p A296676 S:= series(1/(1-arctanh(x)),x,41):
%p A296676 seq(coeff(S,x,j)*j!,j=0..40); # _Robert Israel_, Dec 18 2017
%p A296676 # second Maple program:
%p A296676 a:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
%p A296676       a(n-j)*binomial(n, j)*(j-1)!, 0), j=1..n))
%p A296676     end:
%p A296676 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 22 2021
%t A296676 nmax = 22; CoefficientList[Series[1/(1 - ArcTanh[x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A296676 nmax = 22; CoefficientList[Series[1/(1 + (Log[1 - x] - Log[1 + x])/2), {x, 0, nmax}], x] Range[0, nmax]!
%o A296676 (PARI) x='x+O('x^99); Vec(serlaplace(1/(1+(log(1-x)-log(1+x))/2))) \\ _Altug Alkan_, Dec 18 2017
%Y A296676 Cf. A000246, A000828, A010050, A111594, A191700, A296675.
%K A296676 nonn
%O A296676 0,3
%A A296676 _Ilya Gutkovskiy_, Dec 18 2017