A212913 - OEIS

%I #14 Jul 16 2014 03:33:23

%S 1,1,2,8,44,308,2648,26912,315536,4193744,62302496,1023057536,

%T 18400342208,359733922880,7595810693504,172270928222720,

%U 4176595617747200,107793463235860736,2950683535353324032,85386983313510877184,2604521649171407301632,83519383797513832420352

%N E.g.f. satisfies: A(x) = exp( Integral 1 + x*A(x)^2 dx ), where the constant of integration is zero.

%C Compare to the identities:

%C (1) F(x) = exp( Integral 1 + x*F(x) dx ) when F(x) = 1/(1-x),

%C (2) G(x) = exp( Integral x*G(x)^2 dx ) when G(x) = 1/(1-x^2)^(1/2).

%F E.g.f.: sqrt(2)*exp(x)/sqrt(exp(2*x) - 2*exp(2*x)*x + 1). - _Vaclav Kotesovec_, Jan 05 2014

%F a(n) ~ 2^(n+1) * n^n / (exp(n) * (1+LambertW(exp(-1)))^(n+1)). - _Vaclav Kotesovec_, Jan 05 2014

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 44*x^4/4! + 308*x^5/5! + 2648*x^6/6! +...

%e such that, by definition,

%e log(A(x)) = x + x^2/2! + 4*x^3/3! + 18*x^4/4! + 112*x^5/5! + 880*x^6/6! + 8256*x^7/7! +...

%e Related expansions:

%e x*A(x)^2 = x + 4*x^2/2! + 18*x^3/3! + 112*x^4/4! + 880*x^5/5! + 8256*x^6/6! +...

%e A(x)^2 = 1 + 2*x + 6*x^2/2! + 28*x^3/3! + 176*x^4/4! + 1376*x^5/5! + 12912*x^6/6! +...

%t CoefficientList[Series[Sqrt[2]*E^x/Sqrt[E^(2*x) - 2*E^(2*x)*x + 1], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 05 2014 *)

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(intformal(1+x*A^2)+x*O(x^n)));n!*polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A212914, A245266, A245267.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 30 2012