A177385 E.g.f.: Sum_{n>=0} Product_{k=1..n} sinh(k*x).

1, 1, 4, 37, 616, 16081, 605164, 31011457, 2076192976, 175951716481, 18411425885524, 2331339303739777, 351341718484191736, 62144180030978834881, 12748469150999320273084, 3002313213700366145858497

Offset: 0

Author

Paul D. Hanna, May 15 2010

Keywords

• nonn

Comments

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

. Sum_{n>=0} A002105(n+1)*x^n/n! = Sum_{n>=0} Product_{k=1..n} tanh(k*x).

Limit n->infinity n!*A177386(n) / (2^n*A177385(n)) = 1. - Vaclav Kotesovec, Nov 06 2014

Examples

			E.g.f: A(x) = 1 + x + 4*x^2/2! + 37*x^3/3! + 616*x^4/4! +...
A(x) = 1 + sinh(x) + sinh(x)*sinh(2x) + sinh(x)*sinh(2x)*sinh(3x) + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A249698, A177386, A249564, A249748.

Programs

Mathematica

Table[n!*SeriesCoefficient[Sum[Product[Sinh[k*x],{k,1,j}],{j,0,n}],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 01 2014 *)
nn=20; tab = ConstantArray[0,nn]; tab[[1]] = Series[Sinh[x],{x,0,nn}]; Do[tab[[k]] = Series[tab[[k-1]]*Sinh[k*x],{x,0,nn}],{k,2,nn}]; Flatten[{1,Rest[CoefficientList[Sum[tab[[k]],{k,1,nn}],x] * Range[0,nn]!]}] (* Vaclav Kotesovec, Nov 04 2014 (more efficient) *)

PARI

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

Formula

a(n) ~ c * d^n * (n!)^2, where d = A249748 = 1.04689919262595424111342518325311817976789046475647184115584744582777576864..., c = 0.880333778211172907563073031129920597506533414605109200048966773434616066... . - Vaclav Kotesovec, Nov 04 2014