OFFSET
1,3
FORMULA
E.g.f.: A(x) = sinh(G(x)) where G(x) = Series_Reversion(x - sinh(x)^2) is the e.g.f. of A215093.
a(n) ~ 2^(2*n-3/2) * sqrt(1+1/sqrt(5)) * n^(n-1) / (exp(n) * (1-sqrt(5) + 4*arcsinh(sqrt((sqrt(5)-1)/2)))^(n-1/2)). - Vaclav Kotesovec, Jan 10 2014
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 211*x^5/5! + 2296*x^6/6! +...
A(x) = sinh(G(x)) where G(x) is the e.g.f. of A215093:
G(x) = x + x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! +...
where
A(x)^2/2 = x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! +...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[-x^2/2 + ArcSinh[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 10 2014 *)
PROG
(PARI) {a(n)=n!*polcoeff(sinh(serreverse(x-sinh(x+x*O(x^n))^2/2)), n)}
(PARI) {a(n)=local(A=x); for(i=0, n, A=x + sinh(A)^2/2); n!*polcoeff(sinh(A), n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 02 2012
STATUS
approved