OFFSET
0,3
COMMENTS
Compare to the following cubic modular identity of Jacobi theta functions,
(3*H(q^9)/H(q) - 1)^3 = 9*H(q^3)^4/H(q)^4 - 1,
which holds for H(q) = theta_2(q), H(q) = theta_3(q), and H(q) = theta_4(q) (see Mathworld link).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
Weisstein, Eric W., Modular Equation. From MathWorld -- A Wolfram Web Resource.
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 354*x^4 + 3255*x^5 + 32118*x^6 + 330101*x^7 + 3500381*x^8 + 38002879*x^9 + 420290945*x^10 + ...
RELATED SERIES.
(6*A(x^9)/A(x) - 1)^3 = 125 - 450*x - 1260*x^2 - 12096*x^3 - 68922*x^4 - 634536*x^5 - 6123852*x^6 + ...
126*A(x^3)^4 / A(x)^4 = 126 - 504*x - 1260*x^2 - 12096*x^3 - 68922*x^4 - 634536*x^5 - 6123852*x^6 + ...
so that 126*A(x^3)^4/A(x)^4 - (6*A(x^9)/A(x) - 1)^3 = 1 - 54*x.
PROG
(PARI) {a(n) = my(V=[1], A); for(m=0, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( 126*subst(A, x, x^3)^4/A^4 - 1 + 54*x - (6*subst(A, x, x^9)/A - 1)^3, #V-1)/54 ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2024
STATUS
approved