A233537 E.g.f. satisfies: A'(x) = (1 + x*A(x))*(1 + 2*x*A(x)).

1, 1, 3, 10, 51, 312, 2285, 19776, 193641, 2143872, 26332083, 355752000, 5245533579, 83760362496, 1440626560893, 26546198746368, 521773563403665, 10896758207668224, 240952051977165603, 5624033606823011328, 138178553037552463779, 3564697656160155156480, 96340383688983485779917

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..22.

Formula

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

Limit n->infinity (a(n)/n!)^(1/n) = 1.22846523024810212537857688314... - Vaclav Kotesovec, Dec 20 2013

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

Example

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! +...
where
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! +...

Mathematica

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 *)

Prog

(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)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A048175 A288953 A192482 * A020132 A330042 A241459

Adjacent sequences: A233534 A233535 A233536 * A233538 A233539 A233540

Keyword

nonn

Author

Paul D. Hanna, Dec 15 2013

Status

approved