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A221077 E.g.f.: Sum_{n>=0} tanh(n*x)^n. 8
1, 1, 8, 160, 5888, 345856, 29677568, 3502489600, 544181977088, 107675615297536, 26435436140822528, 7885689342279024640, 2809177794704769548288, 1177952320402008693538816, 574318105367992485583781888, 322156963576521588458420961280, 206009256195720974104252003647488 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4 ...], with an apparent period of 6. - Peter Bala, Jun 01 2022

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..220

FORMULA

E.g.f.: Sum_{n>=0} (exp(2*n*x) - 1)^n / (exp(2*n*x) + 1)^n.

a(n) ~ c * 2^n * (n!)^2 / (sqrt(n) * (log(1+sqrt(2)))^(2*n)), where c = 0.521427744491499132141002572969819345522922990165233786929882335275903215... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

EXAMPLE

E.g.f.: A(x) = 1 + x + 8*x^2/2! + 160*x^3/3! + 5888*x^4/4! + 345856*x^5/5! +...

where

A(x) = 1 + tanh(x) + tanh(2*x)^2 + tanh(3*x)^3 + tanh(4*x)^4 + tanh(5*x)^5 +...

MATHEMATICA

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

Join[{1}, Table[Sum[2^n * k^n * Sum[(-1)^j * Binomial[k, j] * Sum[(-1)^m * Binomial[j + m - 1, m] * StirlingS2[n, m] * m! / 2^m, {m, 1, n}], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jun 01 2022 *)

PROG

(PARI) {a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, tanh(m*X)^m); n!*polcoeff(Egf, n)}

for(n=0, 20, print1(a(n), ", ") )

(PARI) {a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, (exp(2*m*X)-1)^m/(exp(2*m*X)+1)^m); n!*polcoeff(Egf, n)}

for(n=0, 20, print1(a(n), ", ") )

CROSSREFS

Cf. A122399, A195415, A220181, A221078, A221079, A224899, A249489, A245322, A338040.

Sequence in context: A127369 A228700 A036909 * A052140 A219265 A300466

Adjacent sequences:  A221074 A221075 A221076 * A221078 A221079 A221080

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 31 2012

STATUS

approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)