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A328795
Expansion of (chi(x) * chi(-x^3))^2 in powers of x where chi() is a Ramanujan theta function.
3
1, 2, 1, 0, 0, 2, 2, 0, 2, 2, 1, 0, 2, 6, 2, 0, 3, 6, 4, 0, 4, 8, 4, 0, 7, 14, 7, 0, 6, 16, 10, 0, 11, 20, 11, 0, 14, 32, 16, 0, 17, 38, 21, 0, 22, 46, 24, 0, 32, 66, 34, 0, 34, 78, 44, 0, 49, 96, 50, 0, 60, 130, 66, 0, 72, 154, 84, 0, 90, 186, 98, 0, 117, 244
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square of A328802.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328789.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, 0, ...].
G.f.: Product_{k>=1} (1 + x^(2*k-1))^2 * (1 - x^(6*k-3))^2.
a(n) = (-1)^n * A328797(n). a(2*n) = A112206(n).
a(4*n) = A328789(n). a(4*n + 1) = 2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.
EXAMPLE
G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
G.f. = q^-1 + 2*q^2 + q^5 + 2*q^14 + 2*q^17 + 2*q^23 + 2*q^26 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6])^2, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A))^2 / (eta(x+ A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 28 2019
STATUS
approved