A195415 E.g.f.: Sum_{n>=1} tanh(n*x)^n = Sum_{n>=1} a(n)*4^(n-1)*x^n/n!.

1, 2, 10, 92, 1351, 28982, 855100, 33214232, 1642999501, 100843185962, 7520379392890, 669760178257172, 70211429619908851, 8558006664633638942, 1200128210993564085880, 191861070874818576596912, 34685967730611200643509401, 7041037426518318365605795922

Offset

1,2

Comments

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 2, 3, 1, 0, 2, 1, 2, 3, 1, 0, 2, 1, 2, 3, 1, 0, 2, ...], with an apparent period of 6. - Peter Bala, May 29 2022

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..262

Formula

E.g.f.: Sum_{n>=1} ( 1 - 2/(1+exp(2*n*x)) )^n = Sum_{n>=1} a(n)*4^(n-1)/n!.

a(n) ~ c * d^n * n^(2*n + 1/2), where d = 1 / (2 * exp(2) * log(1+sqrt(2))^2) = 0.0871085887239583895519632137900851584739951067757899616766024190... and c = 13.10490857177911562030370300610447966745088413236135355214718... - Vaclav Kotesovec, May 31 2022

a(n) = A221077(n) / 4^(n-1). - Vaclav Kotesovec, Jun 02 2022

Example

E.g.f.: A(x) = x + 8*x^2/2! + 160*x^3/3! + 5888*x^4/4! + 345856*x^5/5! +...
or, equivalently,
A(x) = x + 2*4*x^2/2! + 10*4^2*x^3/3! + 92*4^3*x^4/4! + 1351*4^4*x^5/5! +...
where
A(x) = tanh(x) + tanh(2*x)^2 + tanh(3*x)^3 + tanh(4*x)^4 + tanh(5*x)^5 +...

Maple

seq(coeff(n!/4^(n-1)*series(add(tanh(n*x)^n, n = 1..100), x, 101), x, n), n = 1..100); # Peter Bala, May 29 2022

Mathematica

nmax = 20; Rest[CoefficientList[Series[Sum[Tanh[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! / 4^(Range[0, nmax] - 1)] (* Vaclav Kotesovec, May 31 2022 *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=1, n, tanh(m*X)^m); n!/4^(n-1)*polcoeff(Egf, n)}

Crossrefs

Cf. A122399, A220181, A221077, A221078, A224899, A245322, A338040.

Sequence in context: A108209 A111773 A289020 * A336271 A181084 A063385

Adjacent sequences: A195412 A195413 A195414 * A195416 A195417 A195418

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2011

Status

approved