A296837 -id:A296837 - cp's OEIS frontend

A296838 Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

0, 1, -4, 48, -1186, 50060, -3226206, 294835184, -36270477034, 5779302944436, -1157856177719830, 284876691727454552, -84442374415240892898, 29680054107768128647388, -12205478262363331593956686, 5805823539844285054558025280, -3163004294186696659107788567386

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			log(1 + x*tanh(x/2)) = x^2/2! - 4*x^4/4! + 48*x^6/6! - 1186*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A001469, A003707, A009379, A009399, A110501, A296837.

nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] log(1 + x*tanh(x/2)).

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.306542374188806202228727831923118284841279755635... is the root of the equation r * tan(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

A305922 Expansion of e.g.f. log(1 + 2*x/(exp(x) + 1)).

Original entry on oeis.org

0, 1, -2, 5, -20, 109, -738, 5991, -56760, 614601, -7486670, 101330635, -1508641140, 24503026989, -431137315434, 8169513007215, -165859346028656, 3591802533860497, -82644488286784326, 2013441061219406739, -51777972823724776620, 1401611202556240950645, -39838169568923591411810

Offset: 0

Ilya Gutkovskiy, Jun 14 2018

sign

Logarithmic transform of A036968.

			E.g.f.: A(x) = x - 2*x^2/2! + 5*x^3/3! - 20*x^4/4! + 109*x^5/5! - 738*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..430
N. J. A. Sloane, Transforms
Wikipedia, Genocchi number
Index entries for sequences related to Bernoulli numbers

Cf. A036968, A296837, A296838, A305711.

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(a(j)*j*
      t(n-j)*binomial(n, j), j=1..n-1)/n))(i-> i*euler(i-1, 0))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 04 2018

nmax = 22; CoefficientList[Series[Log[1 + 2 x/(Exp[x] + 1)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = n EulerE[n - 1, 0] - Sum[k Binomial[n, k] (n - k) EulerE[n - k - 1, 0] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]
a[n_] := a[n] = 2 (1 - 2^n) BernoulliB[n] - Sum[k Binomial[n, k] 2 (1 - 2^(n - k)) BernoulliB[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]

cp's OEIS Frontend

A296838 Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

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