A143137 E.g.f. satisfies A(x) = sinh(x + A(x)^2).

1, 2, 13, 140, 2101, 40502, 954073, 26557400, 852911401, 31042592042, 1262704455013, 56767589130980, 2795116027239901, 149590982933830622, 8646295934108179633, 536766403519254357680, 35620604244949591333201

Offset: 1

Author

Paul D. Hanna, Jul 27 2008

Keywords

• nonn

Examples

			A(x) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2101*x^5/5! +...
A(x) = sinh(G(x)) where G(x) is the e.g.f. of A143136:
G(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! +...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..150

Crossrefs

Cf. A143135, A143136.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[-x^2 + ArcSinh[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

PARI

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

PARI

{a(n)=local(A=x);for(i=0,n,A=x + sinh(A)^2);n!*polcoeff(sinh(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 A143136.

a(n) ~ sqrt(1+sqrt(2)) * 2^(n-7/4) * n^(n-1) / (exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014