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A328802
Expansion of chi(x) * chi(-x^3) in powers of x where chi() is a Ramanujan theta function.
2
1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 5, 5, 0, 0, 4, 5, 0, 0, 6, 5, 0, 0, 7, 7, 0, 0, 7, 8, 0, 0, 8, 8, 0, 0, 11, 11, 0, 0, 10, 12, 0, 0, 13, 12, 0, 0, 15
OFFSET
0,25
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square is A328795.
G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A097242.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(1/6) * (eta(q^2)^2 * eta(q^3)) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, 0, ...].
G.f.: Product_{k>=1} (1 + x^(2*k-1)) * (1 - x^(6*k-3)).
a(n) = (-1)^n * A328800. a(4*n) = A097242(n). a(4*n + 1) = A328796(n). a(4*n + 2) = a(4*n + 3) = 0.
EXAMPLE
G.f. = 1 + x + x^5 + x^8 + x^12 + x^13 + x^16 + x^17 + x^20 + ...
G.f. = q^-1 + q^5 + q^29 + q^47 + q^71 + q^77 + q^95 + q^101 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6], {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)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 28 2019
STATUS
approved