A215094 E.g.f. satisfies A(x) = sinh(x + A(x)^2/2).
1, 1, 4, 25, 211, 2296, 30619, 482455, 8768596, 180603511, 4157281129, 105764735440, 2946911156281, 89247262497121, 2919028298593684, 102543779766289705, 3850690682004992491, 153927330069247143976, 6525942204725963508259, 292483420180063453725175
Offset: 1
Keywords
Examples
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! +...
Programs
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Mathematica
Rest[CoefficientList[InverseSeries[Series[-x^2/2 + ArcSinh[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *) -
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), ", "))
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