A368562 - OEIS

%I #10 Dec 30 2023 09:33:09

%S 1,1,4,22,144,1064,8623,75267,698898,6843478,70209485,751028445,

%T 8344927123,96028777007,1141700185040,13996586957076,176645358631455,

%U 2291885063335367,30533724487796062,417268607544901628,5843943369536347505,83810410526002091163,1229907906811449747716

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

%H Paul D. Hanna, <a href="/A368562/b368562.txt">Table of n, a(n) for n = 2..301</a>

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

%F (1) A(x - A(x)) = x^2 - x^3.

%F (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.

%e 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 + ...

%e where A(x - A(x)) = x^2 - x^3.

%e RELATED SERIES.

%e Let B(x) be the g.f. of A190761, then A(x) = B(x)^2 - B(x)^3 where

%e B(x) = x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 592*x^6 + 4670*x^7 + ...

%e Also,

%e A(B(x)) = x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 592*x^6 + 4670*x^7 + ...

%o (PARI) {a(n) = my(A=x^2, B=x);

%o for(i=1,n, A = B^2 - B^3 +x*O(x^n); B = x + subst(A,x,B) ); polcoeff(A,n)}

%o for(n=2,25,print1(a(n),", "))

%Y Cf. A190761.

%K nonn

%O 2,3

%A _Paul D. Hanna_, Dec 30 2023