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A258747 Expansion of chi(-x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions.

%I A258747 #14 Jun 15 2025 06:00:02
%S A258747 1,-1,0,0,0,-1,-2,2,1,0,0,2,0,0,-2,0,1,0,0,0,0,-1,-2,0,2,-2,0,2,0,-2,
%T A258747 0,0,2,-1,0,0,0,0,0,2,3,0,0,0,0,-2,-2,0,0,0,0,0,0,0,-2,2,1,-2,0,2,0,0,
%U A258747 -4,0,2,-1,0,0,0,0,-2,2,0,0,0,2,0,0,0,0,2
%N A258747 Expansion of chi(-x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions.
%C A258747 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258747 G. C. Greubel, <a href="/A258747/b258747.txt">Table of n, a(n) for n = 0..10000</a>
%H A258747 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258747 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258747 Expansion of q^(-1/3) * eta(q) * eta(q^6)^4 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
%F A258747 Euler transform of period 12 sequence [ -1, 0, 0, 0, -1, -3, -1, 0, 0, 0, -1, -2, ...].
%F A258747 G.f.: Product_{k>0} (1 + x^(3*k)) * (1 - x^(6*k))^2 / ( (1 + x^k) * (1 + x^(6*k)) ).
%F A258747 -2 * a(n) = A082564(3*n + 1). a(n) = A129134(3*n + 1).
%F A258747 a(4*n + 3) = 2 * A257402(n-1). a(8*n) = A257398(n). a(8*n + 2) = a(8*n + 4) = a(16*n + 3) = a(16*n + 15) = 0. a(16*n + 7) = 2 * A255318(n). a(16*n + 11) = 2 * A255319(n).
%e A258747 G.f. = 1 - x - x^5 - 2*x^6 + 2*x^7 + x^8 + 2*x^11 - 2*x^14 + x^16 - x^21 + ...
%e A258747 G.f. = q - q^4 - q^16 - 2*q^19 + 2*q^22 + q^25 + 2*q^34 - 2*q^43 + q^49 + ...
%t A258747 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}];
%t A258747 a[ n_] := If[ n < 0, 0, (-1)^Quotient[ 3 n, 2] DivisorSum[ 3 n + 1, KroneckerSymbol[-2, #] &]];
%o A258747 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^4 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
%o A258747 (PARI) {a(n) = if( n<0, 0, (-1)^(3*n\2) * sumdiv(3*n + 1, d, kronecker( -2, d)))};
%Y A258747 Cf. A082564, A129134, A255318, A255319, A257398.
%K A258747 sign
%O A258747 0,7
%A A258747 _Michael Somos_, Jun 09 2015