A235166 E.g.f. satisfies: A'(x) = A(x)^2/A(-x)^2, with A(0)=1.

1, 1, 4, 16, 88, 640, 5440, 54400, 620800, 7966720, 113651200, 1783091200, 30519808000, 565916876800, 11300689100800, 241781039104000, 5517822373888000, 133795711025152000, 3435107208822784000, 93093522064998400000, 2655675672405606400000, 79546285618254315520000

Offset: 0

Author

Paul D. Hanna, Jan 04 2014

Keywords

• nonn

Comments

See comments by Roland Bacher in A098777 which imply that this sequence is related to elliptic functions.

Compare to: G'(x) = G(x)^2/G(-x) dx, which holds when G(x) = 1/(cos(x) - sin(x)), the e.g.f. of A001586 (Springer numbers).

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 88*x^4/4! + 640*x^5/5! +...
Related series.
A(x)^2 = 1 + 2*x + 10*x^2/2! + 56*x^3/3! + 400*x^4/4! + 3440*x^5/5! +...
1/A(x) = 1 - x - 2*x^2/2! + 2*x^3/3! + 16*x^4/4! - 40*x^5/5! - 320*x^6/6! +...+ A098777(n)*x^n/n! +...
		

Links

• Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (first 100 terms from Paul D. Hanna)

Crossrefs

Cf. A001586, A098777.

Programs

Mathematica

kmax = 21;
A[x_] = 1+x; Do[A[x_] = 1+Integrate[A[x]^2/A[-x]^2+O[x]^k, x] // Normal, {k, 1, kmax}];
CoefficientList[A[x], x] Range[0, kmax]! (* Jean-François Alcover, Jul 29 2018 *)

PARI

{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(A^2/subst(A, x,-x +x*O(x^n))^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

PARI

{a(n)=local(A=1); A=1/(1-3*serreverse(intformal(1/(1-9*x^2 +x*O(x^n))^(2/3))))^(1/3); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f.: 1/(1 - 3*Series_Reversion( Integral 1/(1 - 9*x^2)^(2/3) dx ))^(1/3).

E.g.f.: 1/F(x), where F(x) equals the e.g.f. of A098777 (pseudo-factorials).

a(n) ~ 2^(-2/3) * n! * (9*GAMMA(2/3)^3/(2^(2/3)*Pi^2))^(n+1). - Vaclav Kotesovec, Feb 24 2014