A191700 -id:A191700 - cp's OEIS frontend

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

Ilya Gutkovskiy, Dec 18 2017

nonn

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

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

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

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

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]!

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

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

A331617 E.g.f.: exp(1 / (1 - arctan(x)) - 1).

Original entry on oeis.org

1, 1, 3, 11, 49, 265, 1683, 12035, 95169, 832337, 7998467, 83033403, 922112305, 10978263257, 139956480467, 1889161216179, 26798589518593, 401123509624737, 6346168059440515, 105040097140558699, 1805102151607613361, 32421358229074354601

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

sign

a(53) is negative. - Vaclav Kotesovec, Jan 26 2020

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A002019, A110708, A191700, A331610, A331615, A331616, A331618.

nmax = 21; CoefficientList[Series[Exp[1/(1 - ArcTan[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A191700[0] = 1; A191700[n_] := A191700[n] = Sum[Binomial[n, k] If[OddQ[k], (-1)^Boole[IntegerQ[(k + 1)/4]] (k - 1)!, 0] A191700[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A191700[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

seq(n)={Vec(serlaplace(exp(1/(1 - atan(x + O(x*x^n))) - 1)))} \\ Andrew Howroyd, Jan 22 2020

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A191700(k) * a(n-k).

cp's OEIS Frontend

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

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