A392204 G.f. A(x) satisfies A(x - A(x)) = x^2 + x^3.

1, 3, 14, 96, 794, 7450, 76619, 846161, 9901282, 121628550, 1558078533, 20710677095, 284576031153, 4030039735275, 58681107235040, 876865183372364, 13425492683911543, 210340847136666989, 3368501389869814514, 55089013701535075176, 919307894549615256361, 15643110949893733573753

Offset

2,2

Links

Paul D. Hanna, Table of n, a(n) for n = 2..301

Formula

G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.

(1) A(x - A(x)) = x^2 + x^3.

(2) A(x - x^2 - x^3 - A(x)) = (x - A(x))^2 + (x - A(x))^3.

(3) A(x) = B(x)^2 + B(x)^3 such that B(x - A(x)) = x where B(x) = x + A(B(x)) is the g.f. of A211794.

(4) B(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n! is the g.f. of A211794 where B(x - A(x)) = x.

(5) B(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*A(x)^n / n! ) is the g.f. of A211794 where B(x - A(x)) = x.

Example

G.f.: A(x) = x^2 + 3*x^3 + 14*x^4 + 96*x^5 + 794*x^6 + 7450*x^7 + 76619*x^8 + 846161*x^9 + 9901282*x^10 + ...
where A(x - A(x)) = x^2 + x^3.
RELATED SERIES.
Let B(x) satisfy B(x - A(x)) = x, then B(x) is the g.f. of A211794
B(x) = x + x^2 + 5*x^3 + 34*x^4 + 284*x^5 + 2698*x^6 + 28116*x^7 + 314558*x^8 + ... + A211794(n)*x^n + ...
where B(x) = x + A(B(x)).
Also, from B(x) given above, we have the related series
B(x)^2 = x^2 + 2*x^3 + 11*x^4 + 78*x^5 + 661*x^6 + 6304*x^7 + 65624*x^8 + 731640*x^9 + 8627404*x^10 + ...
B(x)^3 = x^3 + 3*x^4 + 18*x^5 + 133*x^6 + 1146*x^7 + 10995*x^8 + 114521*x^9 + 1273878*x^10 + ...
where A(x) = B(x)^2 + B(x)^3.

Prog

(PARI) \\ Using A(x) = B(x)^2 + B(x)^3 where B(x - A(x)) = x.
{a(n) = my(A = x^2, B=x); for(i=1, n, B = serreverse(x - A + x^2*O(x^n));
A = B^2 + B^3); polcoef(A, n)}
for(n=2, 30, print1(a(n), ", "))

Crossrefs

Cf. A211794, A392205, A392203, A213591.

Sequence in context: A395336 A113181 A295105 * A295106 A295107 A295108

Adjacent sequences: A392201 A392202 A392203 * A392205 A392206 A392207

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2026

Status

approved