OFFSET
1,5
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..600
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^(n*(n-1)/2) * (x^n - A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) / (1 - A(x)*x^n)^n.
EXAMPLE
G.f.: A(x) = x + x^2 + x^3 + x^4 + 3*x^5 + 3*x^6 + 8*x^7 + 12*x^8 + 29*x^9 + 48*x^10 + 105*x^11 + 202*x^12 + 420*x^13 + 831*x^14 + 1729*x^15 + ...
where
1 = ... + x^6/(1/x^3 - A(x))^3 + x^3/(1/x^2 - A(x))^2 + x/(1/x - A(x)) + 1 + (x - A(x)) + x*(x^2 - A(x))^2 + x^3*(x^3 - A(x))^3 + x^6*(x^4 - A(x))^4 + ... + x^(n*(n-1)/2)*(x^n - A(x))^n + ...
PROG
(PARI) {a(n) = my(A=[0], M); for(i=1, n, A=concat(A, 0); M = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff(-1 + sum(m=-M, M, x^(m*(m-1)/2) * (x^m - Ser(A))^m ), #A-1)); A[n+1]}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 21 2024
STATUS
approved