A211209 E.g.f.: Sum_{n>=0} cosh(n*x) * tanh(x)^n.

1, 1, 2, 7, 56, 541, 5312, 57667, 742016, 11113081, 184604672, 3330446527, 65282226176, 1390635884821, 31985367007232, 788073542654587, 20690414547009536, 576994154691083761, 17041602531260628992, 531396143657913225847, 17441890388860556804096

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..270

Formula

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

a(n) ~ c * n! / r^n, where r = 0.609377863436... is the root of the equation tanh(r) = exp(-r), c = 0.357427152747929862626923523384943136230795883784223... . In closed form, r = log((1 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3))/3). - Vaclav Kotesovec, Nov 05 2014

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 56*x^4/4! + 541*x^5/5! + 5312*x^6/6! +...
where
A(x) = 1 + cosh(x)*tanh(x) + cosh(2*x)*tanh(x)^2 + cosh(3*x)*tanh(x)^3 + cosh(4*x)*tanh(x)^4 + cosh(5*x)*tanh(x)^5 + cosh(6*x)*tanh(x)^6 +...

Prog

(PARI) {a(n)=n!*polcoeff(sum(k=0, n, cosh(k*x +x*O(x^n)) * tanh(x +x*O(x^n))^k ), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A182055 A358944 A366456 * A363205 A271440 A304984

Adjacent sequences: A211206 A211207 A211208 * A211210 A211211 A211212

Keyword

nonn

Author

Paul D. Hanna, Feb 05 2013

Status

approved