A215094 - OEIS

%I #6 Jan 10 2014 17:03:43

%S 1,1,4,25,211,2296,30619,482455,8768596,180603511,4157281129,

%T 105764735440,2946911156281,89247262497121,2919028298593684,

%U 102543779766289705,3850690682004992491,153927330069247143976,6525942204725963508259,292483420180063453725175

%N E.g.f. satisfies A(x) = sinh(x + A(x)^2/2).

%F E.g.f.: A(x) = sinh(G(x)) where G(x) = Series_Reversion(x - sinh(x)^2) is the e.g.f. of A215093.

%F 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

%e 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! +...

%e A(x) = sinh(G(x)) where G(x) is the e.g.f. of A215093:

%e G(x) = x + x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! +...

%e where

%e A(x)^2/2 = x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! +...

%t Rest[CoefficientList[InverseSeries[Series[-x^2/2 + ArcSinh[x],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 10 2014 *)

%o (PARI) {a(n)=n!*polcoeff(sinh(serreverse(x-sinh(x+x*O(x^n))^2/2)), n)}

%o (PARI) {a(n)=local(A=x); for(i=0, n, A=x + sinh(A)^2/2); n!*polcoeff(sinh(A), n)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A215093, A143137.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Aug 02 2012