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

1, 1, 2, 8, 40, 264, 2048, 18864, 196992, 2330112, 30519552, 440998656, 6940852224, 118501542912, 2177222879232, 42886017982464, 900748014944256, 20107190510714880, 475167358873239552, 11854636521914695680, 311291779253770911744, 8583598112533040332800, 247944624171011289907200

Offset: 0

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

• nonn

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..430

Crossrefs

Cf. A000246, A000828, A010050, A111594, A191700, A296675.

Programs

Maple

S:= series(1/(1-arctanh(x)),x,41):
seq(coeff(S,x,j)*j!,j=0..40); # Robert Israel, Dec 18 2017
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
      a(n-j)*binomial(n, j)*(j-1)!, 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 22 2021

Mathematica

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

PARI

x='x+O('x^99); Vec(serlaplace(1/(1+(log(1-x)-log(1+x))/2))) \\ Altug Alkan, Dec 18 2017

Formula

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

a(n) ~ n! * 4*exp(2) * (exp(2)+1)^(n-1) / (exp(2)-1)^(n+1). - Vaclav Kotesovec, Dec 18 2017

a(n) = Sum_{k=0..n} k! * A111594(n,k). - Seiichi Manyama, Jun 30 2025