OFFSET
0,3
FORMULA
G.f. A(x) satisfies:
(1) A(x) = P(x)/Q(x) where
P(x) = Sum_{n>=0} (n+1) * x^n * A(x)^(3*n) / (1 - x*A(x)^(n+1))^2,
Q(x) = Sum_{n>=0} (n+1) * x^n * A(x)^(2*n) / (1 - x*A(x)^(n+2)).
(2) A(x) = P(x)/Q(x) where
P(x) = Sum_{n>=0} (n+1) * x^n * A(x)^n / (1 - x*A(x)^(n+3))^2,
Q(x) = Sum_{n>=0} x^n * A(x)^(2*n) / (1 - x*A(x)^(n+2))^2.
(3) A(x) = (1/x) * Series_Reversion(x/F(x)), where F(x) is the g.f. of A340942.
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 103*x^4 + 724*x^5 + 5381*x^6 + 41603*x^7 + 331251*x^8 + 2698123*x^9 + 22377963*x^10 + 188354172*x^11 + ...
such that A(x) = P(x)/Q(x) where
P(x) = 1/(1 - x*A(x))^2 + 2*x*A(x)^3/(1 - x*A(x)^2)^2 + 3*x^2*A(x)^6/(1 - x*A(x)^3)^2 + 4*x^3*A(x)^9/(1 - x*A(x)^4)^2 + 5*x^4*A(x)^12/(1 - x*A(x)^5)^2 + ...
Q(x) = 1/(1 - x*A(x)^2) + 2*x*A(x)^2/(1 - x*A(x)^3) + 3*x^2*A(x)^4/(1 - x*A(x)^4) + 4*x^3*A(x)^6(1 - x*A(x)^5) + 5*x^4*A(x)^8/(1 - x*A(x)^6) + ...
equivalently,
P(x) = 1/(1 - x*A(x)^3)^2 + 2*x*A(x)/(1 - x*A(x)^4)^2 + 3*x^2*A(x)^2/(1 - x*A(x)^5)^2 + 4*x^3*A(x)^3/(1 - x*A(x)^6)^2 + 5*x^4*A(x)^4/(1 - x*A(x)^7)^2 + ...
Q(x) = 1/(1 - x*A(x)^2)^2 + x*A(x)^2/(1 - x*A(x)^3)^2 + x^2*A(x)^4/(1 - x*A(x)^4)^2 + x^3*A(x)^6/(1 - x*A(x)^5)^2 + x^4*A(x)^8/(1 - x*A(x)^6)^2 + ...
explicitly,
P(x) = 1 + 4*x + 18*x^2 + 94*x^3 + 565*x^4 + 3770*x^5 + 27031*x^6 + 203724*x^7 + 1591617*x^8 + 12775324*x^9 + 104723264*x^10 + 873036602*x^11 + ...
Q(x) = 1 + 3*x + 12*x^2 + 57*x^3 + 321*x^4 + 2053*x^5 + 14314*x^6 + 105810*x^7 + 815054*x^8 + 6472124*x^9 + 52606288*x^10 + 435562712*x^11 + ...
RELATED SERIES.
A(x) = F(x*A(x)) where F(x) = A(x/F(x)) is the g.f. of A340942:
F(x) = 1 + x + 2*x^2 + 9*x^3 + 46*x^4 + 253*x^5 + 1467*x^6 + 8842*x^7 + 54878*x^8 + 348489*x^9 + 2254007*x^10 + ... + A340942(n)*x^n + ...
PROG
(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^(3*m)/(1 - x*A^(m+1) + x*O(x^n))^2 );
Q = sum(m=0, n, (m+1)*x^m*A^(2*m)/(1 - x*A^(m+2) + x*O(x^n)) );
A = P/Q); polcoeff(H=A, n)}
for(n=0, 20, 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, (m+1)*x^m*A^m/(1 - x*A^(m+3) + x*O(x^n))^2 );
Q = sum(m=0, n, x^m*A^(2*m)/(1 - x*A^(m+2) + x*O(x^n))^2 );
A = P/Q); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 08 2021
STATUS
approved