A296676 - OEIS

%I #13 Jun 22 2021 15:13:14

%S 1,1,2,8,40,264,2048,18864,196992,2330112,30519552,440998656,

%T 6940852224,118501542912,2177222879232,42886017982464,900748014944256,

%U 20107190510714880,475167358873239552,11854636521914695680,311291779253770911744,8583598112533040332800,247944624171011289907200

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

%H Alois P. Heinz, <a href="/A296676/b296676.txt">Table of n, a(n) for n = 0..430</a>

%F E.g.f.: 1/(1 + (log(1 - x) - log(1 + x))/2).

%F a(n) ~ n! * 4*exp(2) * (exp(2)+1)^(n-1) / (exp(2)-1)^(n+1). - _Vaclav Kotesovec_, Dec 18 2017

%e 1/(1 - arctanh(x)) = 1 + x/1! + 2*x^2/2! + 8*x^3/3! + 40*x^4/4! + 264*x^5/5! + ...

%p S:= series(1/(1-arctanh(x)),x,41):

%p seq(coeff(S,x,j)*j!,j=0..40); # _Robert Israel_, Dec 18 2017

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,

%p       a(n-j)*binomial(n, j)*(j-1)!, 0), j=1..n))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 22 2021

%t nmax = 22; CoefficientList[Series[1/(1 - ArcTanh[x]), {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 22; CoefficientList[Series[1/(1 + (Log[1 - x] - Log[1 + x])/2), {x, 0, nmax}], x] Range[0, nmax]!

%o (PARI) x='x+O('x^99); Vec(serlaplace(1/(1+(log(1-x)-log(1+x))/2))) \\ _Altug Alkan_, Dec 18 2017

%Y Cf. A000828, A010050, A191700, A296675.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 18 2017