OFFSET
0,2
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 1 = Sum_{n=-oo..+oo} x^(n+1) * (2 - x^(n+1))^n * A(x)^n.
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / ((1 - 2*x^n)^(n+1) * A(x)^(n+1)).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 78*x^4 + 364*x^5 + 1758*x^6 + 9144*x^7 + 48508*x^8 + 264014*x^9 + 1457624*x^10 + 8158260*x^11 + 46134878*x^12 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^(n+1) * (2 - x^(n+1) +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, (-1)^(n+1) * x^(n^2) / ((1 - 2*x^n +x*O(x^#A) )^(n+1) * Ser(A)^(n+1)) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 22 2022
STATUS
approved