%I #22 Nov 05 2015 18:01:33

%S 1,1,2,7,32,165,920,5451,33932,220127,1479568,10259394,73148986,

%T 534912298,4003678718,30621175937,238991734932,1901317053794,

%U 15403799348464,126986224205129,1064498810334784,9068572636018981,78472353869487618,689421099954372488,6147113115698137860

%N G.f. A(x) satisfies: A(x) = x + B(A(x))^2 such that B(x) = x + A(B(x))^3, where B(x) is the g.f. of A263533.

%H Paul D. Hanna, <a href="/A263532/b263532.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) and B(x) also satisfy:

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

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

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

%F (4) B( x - x^3 - B(x)^2 ) = x - B(x)^2.

%F (5) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) B(x)^(2*n)/n!.

%F (6) B(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/n!.

%F (7) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) B(x)^(2*n)/(n!*x) ).

%F (8) B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/(n!*x) ).

%e G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 165*x^6 + 920*x^7 + 5451*x^8 + 33932*x^9 + 220127*x^10 + 1479568*x^11 + 10259394*x^12 +...

%e such that A(x) = x + B(A(x))^2 and B(x) = x + A(B(x))^3 where

%e B(x) = x + x^3 + 3*x^4 + 12*x^5 + 55*x^6 + 276*x^7 + 1488*x^8 + 8499*x^9 + 50925*x^10 + 317841*x^11 + 2055474*x^12 +...

%e Also

%e A(x) = x + B(x)^2 + d/dx B(x)^4/2! + d^2/dx^2 B(x)^6/3! + d^3/dx^3 B(x)^8/4! +...

%e B(x) = x + A(x)^3 + d/dx A(x)^6/2! + d^2/dx^2 A(x)^9/3! + d^3/dx^3 A(x)^12/4! +...

%e RELATED SERIES.

%e A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 34*x^6 + 156*x^7 + 804*x^8 + 4469*x^9 + 26307*x^10 + 162306*x^11 + 1042111*x^12 +...

%e B(x)^2 = x^2 + 2*x^4 + 6*x^5 + 25*x^6 + 116*x^7 + 585*x^8 + 3158*x^9 + 18024*x^10 + 107802*x^11 + 671257*x^12 +...

%e Also

%e A(B(x)) = x + x^2 + 3*x^3 + 12*x^4 + 56*x^5 + 291*x^6 + 1634*x^7 + 9738*x^8 + 60887*x^9 + 396259*x^10 + 2669199*x^11 + 18531931*x^12 +...

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

%e And

%e B(A(x)) = x + x^2 + 3*x^3 + 13*x^4 + 65*x^5 + 356*x^6 + 2090*x^7 + 12963*x^8 + 84090*x^9 + 566495*x^10 + 3943195*x^11 + 28252008*x^12 +...

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

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

%o for(n=1,30,print1(a(n),", "))

%o (PARI) {a(n) = my(A=x,B=x); for(i=1,n, A = serreverse(x - B^2 +x*O(x^n)); B = serreverse(x - A^3);); polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A263533.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Nov 03 2015