OFFSET
0,2
COMMENTS
Related formal power series identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n)^n.
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) 2 = Sum_{n=-oo..+oo} x^(2*n) * (1 - x^n)^(2*n) * A(x)^n.
(2) 2 = Sum_{n=-oo..+oo} x^(2*n*(n-1)) / ((1 - x^n)^(2*n) * A(x)^n).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 4*x^2 + 6*x^3 + 12*x^4 + 18*x^5 + 52*x^6 + 92*x^7 + 278*x^8 + 606*x^9 + 1736*x^10 + 4378*x^11 + 11974*x^12 + ...
such that A(x) satisfies the doubly infinite series
2 = Sum_{n=-oo..+oo} x^(2*n) * (1 - x^n)^(2*n) * A(x)^n
which can be split into infinite series P and Q given by
2 = P + Q ;
P = 1/((1-x)^2*A(x)) + x^4/((1-x^2)^4*A(x)^2) + x^12/((1-x^3)^6*A(x)^3) + ... + x^(2*n*(n-1))/((1 - x^n)^(2*n)*A(x)^n) + ... ;
Q = 1 + x^2*(1-x)^2*A(x) + x^4*(1-x^2)^4*A(x)^2 + x^6*(1-x^3)^6*A(x)^3 + ... + x^(2*n)*(1 - x^n)^(2*n)*A(x)^n + ...
Explicitly,
P = 1 - x^2 - 2*x^4 - 4*x^5 - 13*x^6 - 18*x^7 - 75*x^8 - 126*x^9 - 475*x^10 - 974*x^11 - 2991*x^12 - 7184*x^13 - 20257*x^14 - ...
Q = 1 + x^2 + 2*x^4 + 4*x^5 + 13*x^6 + 18*x^7 + 75*x^8 + 126*x^9 + 475*x^10 + 974*x^11 + 2991*x^12 + 7184*x^13 + 20257*x^14 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(sum(m=-#A, #A, x^(2*m) * (1 - x^m +x*O(x^#A))^(2*m) * Ser(A)^m), #A-1); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 17 2022
STATUS
approved