A233537 - OEIS

%I #15 Dec 20 2013 18:36:17

%S 1,1,3,10,51,312,2285,19776,193641,2143872,26332083,355752000,

%T 5245533579,83760362496,1440626560893,26546198746368,521773563403665,

%U 10896758207668224,240952051977165603,5624033606823011328,138178553037552463779,3564697656160155156480,96340383688983485779917

%N E.g.f. satisfies: A'(x) = (1 + x*A(x))*(1 + 2*x*A(x)).

%C Compare to: G'(x) = (1 + x*G(x))^2 holds when G(x) = 1/(1-x).

%F E.g.f.: 1/(-x + 1/(x + 2/(exp(x^2/2)*(2 + sqrt(2*Pi)*erf(x/sqrt(2)))))). - _Vaclav Kotesovec_, Dec 20 2013

%F Limit n->infinity (a(n)/n!)^(1/n) = 1.22846523024810212537857688314... - _Vaclav Kotesovec_, Dec 20 2013

%F a(n) ~ n! * c * (1/r)^n, where r = 0.8140238529974828444777... is the root of the equation erf(r/sqrt(2)) = sqrt(2/Pi)*(r*exp(-r^2/2)/(1-r^2)-1) and c = 0.9269549143870045466948... - _Vaclav Kotesovec_, Dec 20 2013

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 51*x^4/4! + 312*x^5/5! + 2285*x^6/6! +...

%e where

%e A'(x) = 1 + 3*x*A(x) + 2*x^2*A(x)^2 = 1 + 3*x + 10*x^2/2! + 51*x^3/3! + 312*x^4/4! + 2285*x^5/5! +...

%t CoefficientList[Series[1/(-x + 1/(x + 2/(E^(x^2/2)*(2 + Sqrt[2*Pi]* Erf[x/Sqrt[2]])))), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Dec 20 2013 *)

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

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 15 2013