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A260308 Expansion of psi(x) * phi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A260308 #9 Feb 16 2025 08:33:26
%S A260308 1,1,0,3,2,0,3,0,0,2,1,0,2,4,0,3,0,0,4,0,0,1,2,0,2,0,0,4,3,0,2,2,0,4,
%T A260308 0,0,1,2,0,2,2,0,2,0,0,1,0,0,8,2,0,2,0,0,2,3,0,2,4,0,0,0,0,4,0,0,1,2,
%U A260308 0,4,0,0,2,0,0,2,4,0,5,0,0,4,2,0,2,2,0
%N A260308 Expansion of psi(x) * phi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A260308 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260308 G. C. Greubel, <a href="/A260308/b260308.txt">Table of n, a(n) for n = 0..1000</a>
%H A260308 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260308 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260308 Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^6)^5 / (eta(q) * eta(q^3)^2 * eta(q^12)^2) in powers of q.
%F A260308 Euler transform of period 12 sequence [ 1, -1, 3, -1, 1, -4, 1, -1, 3, -1, 1, -2, ...].
%F A260308 a(n) = A259668(2*n) = A128580(4*n) = A129402(4*n) = A134177(4*n) = A190615(4*n) = A115660(8*n + 1) = A128581(8*n + 1) = A192013(8*n + 1).
%e A260308 G.f. = 1 + x + 3*x^3 + 2*x^4 + 3*x^6 + 2*x^9 + x^10 + 2*x^12 + 4*x^13 + ...
%e A260308 G.f. = q + q^9 + 3*q^25 + 2*q^33 + 3*q^49 + 2*q^73 + q^81 + 2*q^97 + ...
%t A260308 a[ n_] := If[ n < 0, 0, DivisorSum[ 8 n + 1, KroneckerSymbol[ -6, #] &]];
%t A260308 a[ n_] := If[ n < 0, 0, Times @@ (Which[ # <= 3, Mod[#, 2], Mod[#, 24] > 12, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 3, #]^#2] & @@@ FactorInteger @ (8 n + 1))];
%t A260308 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];
%o A260308 (PARI) {a(n) = if( n<0, 0, sumdiv( 8*n + 1, d, kronecker( -6, d)))};
%o A260308 (PARI) {a(n) = my(A, p, e); if( n<0, 0, factor(8*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, 1, p%24>12, !(e%2), (e+1) * kronecker(3, p)^e)))};
%o A260308 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};
%Y A260308 Cf. A115660, A128580, A128581, A129402, A134177, A190615, A192013, A259668.
%K A260308 nonn
%O A260308 0,4
%A A260308 _Michael Somos_, Jul 22 2015