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A258764 Expansion of chi(-x^2) * psi(-x^3)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

%I A258764 #11 Feb 16 2025 08:33:25
%S A258764 1,0,-1,-2,0,2,0,0,0,0,-1,0,2,2,-2,0,1,0,0,-2,0,0,-2,0,0,0,0,-2,2,2,0,
%T A258764 0,1,0,0,-2,0,2,0,0,0,0,-1,-2,2,2,0,0,2,0,-2,0,0,0,-2,0,0,0,-2,-2,0,0,
%U A258764 0,0,2,0,-1,0,0,4,0,0,0,0,0,-2,0,2,-2,0,3
%N A258764 Expansion of chi(-x^2) * psi(-x^3)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
%C A258764 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258764 G. C. Greubel, <a href="/A258764/b258764.txt">Table of n, a(n) for n = 0..10000</a>
%H A258764 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258764 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258764 Expansion of q^(-2/3) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2 / (eta(q^4) * eta(q^6)^2) in powers of q.
%F A258764 Euler transform of period 12 sequence [ 0, -1, -2, 0, 0, -1, 0, 0, -2, -1, 0, -2, ...].
%F A258764 G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(3*k))^2 * (1 - x^(2*k) + x^(4*k))^2.
%F A258764 a(n) = A129134(3*n + 2). -2 * a(n) = A082564(3*n + 2).
%F A258764 a(4*n) = A257399(n). a(8*n + 3) = -2 * A255318(n). a(8*n + 5) = 2 * A255319(n). a(8*n + 6) = -2 * A257402(n-1). a(16*n) = A257398(n). a(16*n + 2) = - A257399(n). a(16*n + 12) = 2 * A255317(n).
%F A258764 a(8*n + 1) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.
%e A258764 G.f. = 1 - x^2 - 2*x^3 + 2*x^5 - x^10 + 2*x^12 + 2*x^13 - 2*x^14 + x^16 + ...
%e A258764 G.f. = q^2 - q^8 - 2*q^11 + 2*q^17 - q^32 + 2*q^38 + 2*q^41 - 2*q^44 + ...
%t A258764 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4] QPochhammer[ x^3]^2 / QPochhammer[ x^6, x^12]^2, {x, 0, n}];
%t A258764 a[ n_] := If[ n < 0, 0, (-1)^Quotient[ n, 2] DivisorSum[ 3 n + 2, KroneckerSymbol[-2, #] &]];
%o A258764 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x^4 + A) * eta(x^6 + A)^2), n))};
%o A258764 (PARI) {a(n) = if( n<0, 0, (-1)^(n\2) * sumdiv(3*n + 2, d, kronecker( -2, d)))};
%Y A258764 Cf. A082564, A129134, A255317, A255318, A255319, A257399, A257402.
%K A258764 sign
%O A258764 0,4
%A A258764 _Michael Somos_, Jun 09 2015