A229619 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)).

1, 1, 4, 27, 248, 2822, 37820, 578915, 9918924, 187558638, 3873705128, 86692262942, 2089070253556, 53925007946392, 1484529898970648, 43421639185592359, 1344923240469786704, 43981996770022295714, 1514531024603022580980, 54783958839510354056018, 2077007174758224026216216

Offset: 1

Author

Paul D. Hanna, Sep 26 2013

Keywords

• nonn

Comments

a(n) = A360950(n-1)/n for n >= 1. [corrected by Vaclav Kotesovec, Feb 27 2023]

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2822*x^6 + ...
By definition, A(x - x^2*A'(x)) = x, where
A'(x) = 1 + 2*x + 12*x^2 + 108*x^3 + 1240*x^4 + 16932*x^5 + 264740*x^6 + 4631320*x^7 + ... + A360950(n)*x^n + ...
Related expansions.
A'(A(x)) = 1 + 2*x + 14*x^2 + 140*x^3 + 1726*x^4 + 24752*x^5 + ...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 62*x^5 + 566*x^6 + 6356*x^7 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A360950.

Programs

PARI

{a(n)=local(A=x+x^2);for(i=1,n,A=serreverse(x-x^2*A'+x*O(x^n)));polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = x + A(x)^2 * A'(A(x)).

a(n) ~ c * n! * n^(3*LambertW(1) - 2 + 1/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.109236306585641816289... - Vaclav Kotesovec, Feb 27 2023