%I #6 Feb 11 2021 10:31:28

%S 1,1,2,6,22,91,407,1921,9429,47683,246901,1303346,6992087,38031159,

%T 209348857,1164616227,6540112446,37040976542,211423703225,

%U 1215450693258,7034282005208,40966313765380,240003678300088,1414101405300096

%N G.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (n+1)*x^n*A(x)^n/(1 - x*A(x)^(n+1)) and Q(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(n+1)).

%F G.f. A(x) satisfies:

%F (1) A(x) = P(x)/Q(x) where

%F P(x) = Sum_{n>=0} (n+1) * x^n * A(x)^n / (1 - x*A(x)^(n+1)) and

%F Q(x) = Sum_{n>=0} x^n * A(x)^n / (1 - x*A(x)^(n+1)).

%F (2) A(x) = P(x)/Q(x) where

%F P(x) = Sum_{n>=0} x^n * A(x)^n / (1 - x*A(x)^(n+1))^2 and

%F Q(x) = Sum_{n>=0} x^n * A(x)^n / (1 - x*A(x)^(n+1)).

%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + 407*x^6 + 1921*x^7 + 9429*x^8 + 47683*x^9 + 246901*x^10 + 1303346*x^11 + 6992087*x^12 + ...

%e such that A(x) = P(x)/Q(x) where

%e P(x) = 1/(1-x*A(x)) + 2*x*A(x)/(1 - x*A(x)^2) + 3*x^2*A(x)^2/(1 - x*A(x)^3) + 4*x^3*A(x)^3/(1 - x*A(x)^4) + 5*x^4*A(x)^4/(1 - x*A(x)^5) + ...

%e Q(x) = 1/(1-x*A(x)) + x*A(x)/(1 - x*A(x)^2) + x^2*A(x)^2/(1 - x*A(x)^3) + x^3*A(x)^3/(1 - x*A(x)^4) + x^4*A(x)^4/(1 - x*A(x)^5) + ...

%e also

%e P(x) = 1/(1-x*A(x))^2 + x*A(x)/(1 - x*A(x)^2)^2 + x^2*A(x)^2/(1 - x*A(x)^3)^2 + x^3*A(x)^3/(1 - x*A(x)^4)^2 + x^4*A(x)^4/(1 - x*A(x)^5)^2 + ...

%e explicitly,

%e P(x) = 1 + 3*x + 9*x^2 + 30*x^3 + 111*x^4 + 448*x^5 + 1937*x^6 + 8837*x^7 + 42046*x^8 + 206821*x^9 + 1044977*x^10 + 5397263*x^11 + ...

%e Q(x) = 1 + 2*x + 5*x^2 + 15*x^3 + 52*x^4 + 201*x^5 + 843*x^6 + 3760*x^7 + 17579*x^8 + 85259*x^9 + 425772*x^10 + 2177369*x^11 + ...

%o (PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1); for(i=0, n,

%o P = sum(m=0, n, (m+1)*x^m*A^m/(1 - x*A^(m+1) + x*O(x^n)) );

%o Q = sum(m=0, n, x^m*A^m/(1 - x*A^(m+1) + x*O(x^n)) );

%o A = P/Q); polcoeff(H=A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1); for(i=0, n,

%o P = sum(m=0, n, x^m*A^m/(1 - x*A^(m+1) + x*O(x^n))^2 );

%o Q = sum(m=0, n, x^m*A^m/(1 - x*A^(m+1) + x*O(x^n)) );

%o A = P/Q); polcoeff(H=A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A341342.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 10 2021