OFFSET
0,2
COMMENTS
The g.f. A(x) of this sequence is motivated by the following identity:
Sum_{n>=0} C(t+n-1,n) * p^n/(1 - q*r^n)^s = Sum_{n>=0} C(s+n-1,n) * q^n/(1 - p*r^n)^t ;
here, p = x, q = x*A(x), r = x, s = 2, and t = 3.
FORMULA
G.f. A(x) satisfies the following relations.
(1) A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x))^3.
(2) A(x) = Sum_{n>=0} (n+1)*(n+2)/2 * x^n * A(x)^n / (1 - x^(n+1))^2.
EXAMPLE
G.f.: A(x) = 1 + 5*x + 24*x^2 + 152*x^3 + 1094*x^4 + 8508*x^5 + 69565*x^6 + 588469*x^7 + 5106516*x^8 + 45199827*x^9 + 406485567*x^10 + ...
where
A(x) = 1/(1 - x*A(x))^3 + 2*x/(1 - x^2*A(x))^3 + 3*x^2/(1 - x^3*A(x))^3 + 4*x^3/(1 - x^4*A(x))^3 + 5*x^4/(1 - x^5*A(x))^3 + ...
also
A(x) = 1/(1 - x)^2 + 3*x*A(x)/(1 - x^2)^2 + 6*x^2*A(x)^2/(1 - x^3)^2 + 10*x^3*A(x)^3/(1 - x^4)^2 + 15*x^4*A(x)^4/(1 - x^5)^2 + ...
PROG
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, (m+1) * x^m / (1 - x^(m+1)*A +x*O(x^n))^3 )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, (m+1)*(m+2)/2 * x^m * A^m / (1 - x^(m+1) +x*O(x^n))^2 )); ; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 07 2021
STATUS
approved