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A260158 Expansion of psi(x)^4 * psi(-x^3) / f(x) in powers of x where psi, f() are Ramanujan theta functions.

%I A260158 #18 Feb 16 2025 08:33:26
%S A260158 1,3,4,6,7,6,10,12,13,15,14,18,18,21,22,18,25,27,28,24,26,33,34,42,37,
%T A260158 30,36,42,43,45,38,48,49,42,54,42,56,57,58,60,43,63,64,66,67,63,70,60,
%U A260158 73,84,62,78,79,72,72,66,90,87,88,90,74,78,98,96,97,78
%N A260158 Expansion of psi(x)^4 * psi(-x^3) / f(x) in powers of x where psi, f() are Ramanujan theta functions.
%C A260158 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260158 G. C. Greubel, <a href="/A260158/b260158.txt">Table of n, a(n) for n = 0..10000</a>
%H A260158 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260158 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260158 Expansion of q^(-5/6) * eta(q^2)^5 * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q)^3 * eta(q^6)) in powers of q.
%F A260158 Euler transform of period 12 sequence [ 3, -2, 2, -3, 3, -2, 3, -3, 2, -2, 3, -4, ...].
%F A260158 4 * a(n) = A260109(3*n + 2) = A124815(6*n + 5).
%F A260158 a(2*n + 1) = 3 * A260295(n).
%e A260158 G.f. = 1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 6*x^5 + 10*x^6 + 12*x^7 + 13*x^8 + ...
%e A260158 G.f. = q^7 + 3*q^23 + 4*q^39 + 6*q^55 + 7*q^71 + 6*q^87 + 10*q^103 + 12*q^119 + ...
%t A260158 a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 5}, DivisorSum[ m, m/# KroneckerSymbol[ 12, #]&] / 4]];
%t A260158 a[ n_] := SeriesCoefficient[ 2^(-9/2) x^(-7/8) EllipticTheta[ 2, 0, x^(1/2)]^4 EllipticTheta[ 2, Pi/4, x^(3/2)] / QPochhammer[ -x], {x, 0, n}];
%o A260158 (PARI) {a(n) = my(m); if( n<0, 0, m = 6*n + 5; sumdiv( m, d, m/d * kronecker( 12, d)) / 4)};
%o A260158 (PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^6 + A)), n))};
%Y A260158 Cf. A124815, A260109, A260295.
%K A260158 nonn
%O A260158 0,2
%A A260158 _Michael Somos_, Nov 09 2015