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A129451 Expansion of f(-x, -x^3) f(-x, x^2) in powers of x where f(, ) is Ramanujan's general theta function.

%I A129451 #12 Feb 16 2025 08:33:05
%S A129451 1,-2,2,-2,1,-2,2,-2,3,0,2,-2,2,-2,0,-4,2,-2,2,0,1,-2,4,-2,0,-2,2,-2,
%T A129451 3,-2,2,0,2,-2,0,-2,4,-2,2,0,2,-4,0,-4,0,-2,2,-2,1,0,4,-2,2,0,2,-2,2,
%U A129451 -4,2,0,3,-2,2,-2,0,0,2,-4,2,0,2,-4,2,-2,0,0,2
%N A129451 Expansion of f(-x, -x^3) f(-x, x^2) in powers of x where f(, ) is Ramanujan's general theta function.
%C A129451 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)
%H A129451 G. C. Greubel, <a href="/A129451/b129451.txt">Table of n, a(n) for n = 0..10000</a>
%H A129451 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A129451 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A129451 Expansion of psi(-x) * phi(x^3) / chi(x) = f(-x^2)^3 * phi(x^3) / f(x)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
%F A129451 Expansion of q^(-1/6) * eta(q)^2 * eta(q^4)^2 * eta(q^6)^5 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^2) in powers of q.
%F A129451 Euler transform of period 12 sequence [ -2, 1, 0, -1, -2, -2, -2, -1, 0, 1, -2, -2, ...].
%F A129451 a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 5, 11 (mod 12), b(p^e) = e+1 if p == 1 (mod 12), b(p^e) = (-1)^e* (e+1) if p == 7 (mod 12).
%F A129451 a(n) = A129449(3*n).
%e A129451 G.f. = 1 - 2*x + 2*x^2 - 2*x^3 + x^4 - 2*x^5 + 2*x^6 - 2*x^7 + 3*x^8 + 2*x^10 + ...
%e A129451 G.f. = q - 2*q^7 + 2*q^13 - 2*q^19 + q^25 - 2*q^31 + 2*q^37 - 2*q^43 + 3*q^49 + ...
%t A129451 a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 1}, DivisorSum[ m, KroneckerSymbol[ -4, #] KroneckerSymbol[ 12, m/#] &]]]; (* _Michael Somos_, Nov 11 2015 *)
%t A129451 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] QPochhammer[ x^2]^3 / QPochhammer[ -x]^2, {x, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)
%t A129451 a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-1/8) EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 3, 0, x^3] QPochhammer[ x, -x], {x, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)
%o A129451 (PARI) {a(n) = if( n<0, 0, n = 6*n + 1; sumdiv(n, d, kronecker( -4, d) * kronecker( 12, n/d)))};
%o A129451 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};
%Y A129451 Cf. A129449.
%K A129451 sign
%O A129451 0,2
%A A129451 _Michael Somos_, Apr 16 2007