A368562 Expansion of g.f. A(x) satisfying A(x - A(x)) = x^2 - x^3.

1, 1, 4, 22, 144, 1064, 8623, 75267, 698898, 6843478, 70209485, 751028445, 8344927123, 96028777007, 1141700185040, 13996586957076, 176645358631455, 2291885063335367, 30533724487796062, 417268607544901628, 5843943369536347505, 83810410526002091163, 1229907906811449747716

Offset: 2

Author

Paul D. Hanna, Dec 30 2023

Keywords

• nonn

Examples

			G.f.: A(x) = x^2 + x^3 + 4*x^4 + 22*x^5 + 144*x^6 + 1064*x^7 + 8623*x^8 + 75267*x^9 + 698898*x^10 + 6843478*x^11 + 70209485*x^12 + ...
where A(x - A(x)) = x^2 - x^3.
RELATED SERIES.
Let B(x) be the g.f. of A190761, then A(x) = B(x)^2 - B(x)^3 where
B(x) = x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 592*x^6 + 4670*x^7 + ...
Also,
A(B(x)) = x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 592*x^6 + 4670*x^7 + ...
		

Links

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

Crossrefs

Cf. A190761.

Programs

PARI

{a(n) = my(A=x^2, B=x);
for(i=1,n, A = B^2 - B^3 +x*O(x^n); B = x + subst(A,x,B) ); polcoeff(A,n)}
for(n=2,25,print1(a(n),", "))

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) = B(x)^2 - B(x)^3 where B(x) = x + A(B(x)) is the g.f. of A190761 and B(x - A(x)) = x.