OFFSET
0,3
COMMENTS
Note that 1 + 4*Sum_{n>=1} (-1)^n * x^(2*n-1)/(1 + x^(2*n-1)) = theta_4(x)^2, where theta_4(x) = 1 + 2*Sum_{n>=1} (-x)^(n^2) is Jacobi's elliptic theta function.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..900
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 13*x^5 + 26*x^6 + 50*x^7 + 96*x^8 + 190*x^9 + 377*x^10 + 747*x^11 + 1494*x^12 + ...
such that
x = x*A(x)/(1 + x*A(x)) - x^3*A(x)/(1 + x^3*A(x)) + x^5*A(x)/(1 + x^5*A(x)) - x^7*A(x)/(1 + x^7*A(x)) + x^9*A(x)/(1 + x^9*A(x)) -+ ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec( sum(m=1, #A, (-1)^m*x^(2*m-1)*Ser(A)/(1+x^(2*m-1)*Ser(A) )) )[#A] ); A[n+1]}
for(n=0, 60, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 20 2018
STATUS
approved