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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)).

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

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

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)^23) + 4*x^3*A(x)^3/(1 - x*A(x)^34) + 5*x^4*A(x)^4/(1 - x*A(x)^4) + 6*x^5*A(x)^5/(1 - x*A(x)^5) + ...

Q(x) = 1/(1-x*A(x)) + x*A(x)/(1 - x*A(x)^2) + x^2*A(x)^2/(1 - x*A(x)^23) + x^3*A(x)^3/(1 - x*A(x)^34) + x^4*A(x)^4/(1 - x*A(x)^4) + x^5*A(x)^5/(1 - x*A(x)^4) + ...

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)^23)^2 + x^3*A(x)^3/(1 - x*A(x)^34)^2 + x^4*A(x)^4/(1 - x*A(x)^4)^2 + x^5*A(x)^5/(1 - x*A(x)^5)^2 + ...

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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)).

1, 1, 2, 6, 22, 91, 407, 1921, 9429, 47683, 246901, 1303346, 6992087, 38031159, 209348857, 1164616227, 6540112446, 37040976542, 211423703225, 1215450693258, 7034282005208, 40966313765380, 240003678300088, 1414101405300096

0,3

G.f. A(x) satisfies:

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

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

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

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

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

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

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 + ...

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

P(x) = 1/(1-x) + 2*x*A(x)/(1 - x*A(x)) + 3*x^2*A(x)^2/(1 - x*A(x)^2) + 4*x^3*A(x)^3/(1 - x*A(x)^3) + 5*x^4*A(x)^4/(1 - x*A(x)^4) + 6*x^5*A(x)^5/(1 - x*A(x)^5) + ...

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

also

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

explicitly,

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 + ...

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 + ...

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

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

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

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

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

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

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

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

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

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

Cf. A341342.

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Paul D. Hanna, Feb 10 2021

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allocated for Paul D. Hanna

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