A187802 E.g.f.: Sum_{n>=0} Product_{k=1..n} tanh(n*k*x).

1, 1, 16, 970, 146176, 44183536, 23478931456, 20054284098640, 25800626187206656, 47592874959658936576, 121099500781576410628096, 411996060596290629454466560, 1826628916277875316651443879936, 10329535274999799577516027932553216, 73156530986984637348101331408897703936

Offset: 0

Author

Paul D. Hanna, Jan 06 2013

Keywords

• nonn

Comments

Compare to the e.g.f. of A002105, the reduced tangent numbers:

Sum_{n>=0} Product_{k=1..n} tanh(k*x).

Examples

			E.g.f.: A(x) = 1 + x + 16*x^2/2! + 970*x^3/3! + 146176*x^4/4! +...
where
A(x) = 1 + tanh(x) + tanh(2*1*x)*tanh(2*2*x) + tanh(3*1*x)*tanh(3*2*x)*tanh(3*3*x) + tanh(4*1*x)*tanh(4*2*x)*tanh(4*3*x)*tanh(4*4*x) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..140

Crossrefs

Cf. A002105.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[Sum[Product[Tanh[n*k*x],{k,n}],{n,0,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 26 2024 *)

PARI

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, prod(k=1, m, tanh(m*k*X))); n!*polcoeff(Egf, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 2.67441747301630303932685879..., c = 0.4405132627693901422580367... . - Vaclav Kotesovec, Nov 02 2014