A249748 -id:A249748 - cp's OEIS frontend

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

Paul D. Hanna, May 15 2010

nonn

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

			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) + ...

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

Cf. A249698, A177386, A249564, A249748.

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) *)

{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)}

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

A177386 O.g.f.: Sum_{n>=0} Product_{k=1..n} sinh(k*arcsinh(2x)).

Original entry on oeis.org

1, 2, 8, 48, 400, 4192, 52720, 773536, 12970016, 244625088, 5125896112, 118137655840, 2970016739552, 80883641686848, 2372035401856352, 74528583049288768, 2497667361588205632, 88932255196677684608

Offset: 0

Paul D. Hanna, May 15 2010

nonn

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

			O.g.f.: A(x) = 1 + 2*x + 8*x^2 + 48*x^3 + 400*x^4 + 4192*x^5 + ...
Let G(x) be the e.g.f. of A177385:
G(x) = 1 + x + 4*x^2/2! + 37*x^3/3! + 616*x^4/4! + 16081*x^5/5! + ...
then A(x) = G(arcsinh(2x)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A177385, A249748.

{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,sinh(k*asinh(2*X))));polcoeff(Egf,n)}

O.g.f.: A(x) = G(arcsinh(2x)) where G(x) = e.g.f. of A177385.

a(n) ~ c * d^n * n!, where d = 2*A249748 = 2.0937983852519084822268503..., c = 0.880333778211172907563073... (constant c is same as for A177385). - Vaclav Kotesovec, Nov 06 2014

cp's OEIS Frontend

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

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