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A177385
E.g.f.: Sum_{n>=0} Product_{k=1..n} sinh(k*x).
9
1, 1, 4, 37, 616, 16081, 605164, 31011457, 2076192976, 175951716481, 18411425885524, 2331339303739777, 351341718484191736, 62144180030978834881, 12748469150999320273084, 3002313213700366145858497
OFFSET
0,3
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
LINKS
FORMULA
a(n) ~ c * d^n * (n!)^2, where d = A249748 = 1.04689919262595424111342518325311817976789046475647184115584744582777576864..., c = 0.880333778211172907563073031129920597506533414605109200048966773434616066... . - Vaclav Kotesovec, Nov 04 2014
EXAMPLE
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) + ...
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) *)
PROG
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 15 2010
STATUS
approved