OFFSET
0,2
COMMENTS
In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then
(C.1) G(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * p^(n-k) * q^k * G(x)^k ).
(C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).
(C.3) x = (sqrt((p - q*y)^2 + 4*p*q*y^2) - (p + q*y))/(2*p*q*y^2), where y = G(x).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = (1 + 2*x*A(x)) * (1 + 3*x*A(x)^2).
(2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 2^(n-k) * 3^k * A(x)^k ).
(3) A(x) = (1/x) * Series_Reversion( x*(1 - 3*x - 6*x^2)/(1 + 2*x) ).
(4) A(x) = Sum_{n>=0} A154964(n+1) * x^n * A(x)^n, where g.f. of A154964 = (1 - 2*x - 4*x^2)/(1 - 3*x - 6*x^2).
(5) x = (sqrt(33*A(x)^2 - 12*A(x) + 4) - (2 + 3*A(x)))/(12*A(x)^2).
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - Seiichi Manyama, Sep 08 2024
EXAMPLE
G.f. A(x) = 1 + 5*x + 46*x^2 + 533*x^3 + 6922*x^4 + 96338*x^5 + 1404796*x^6 + 21184229*x^7 + 327659314*x^8 + 5169425894*x^9 + 82866843652*x^10 + ...
where A(x) = (1 + 2*x*A(x)) * (1 + 3*x*A(x)^2).
RELATED SERIES.
Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then
B(x) = 1 + 5*x + 21*x^2 + 93*x^3 + 405*x^4 + 1773*x^5 + 7749*x^6 + 33885*x^7 + ... + A154964(n+1)*x^n + ...
where B(x) = (1 + 2*x)/(1 - 3*x - 6*x^2).
PROG
(PARI) {a(n) = my(A=1+x); for(i=1, n, A=(1 + 2*x*A)*(1 + 3*x*(A+x*O(x^n))^2)); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=polcoef( (1/x)*serreverse( x*(1 - 3*x - 6*x^2)/(1 + 2*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2 * 2^(m-j) * 3^j * A^j)*x^m/m))); polcoef(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2024
STATUS
approved