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%I #12 Mar 12 2021 22:24:48
%S 1,6,14,18,21,30,38,42,43,48,62,66,74,78,64,84,98,102,110,96,133,126,
%T 108,138,112,150,158,162,183,126,182,192,194,198,160,210,180,222,230,
%U 192,242,252,288,228,208,270,278,282,273,240,252,306,314,336,294,330
%N Expansion of f(x)^4 * phi(-x^3) / phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A260114/b260114.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1/6) * eta(q^2)^13 * eta(q^3)^2 / (eta(q)^6 * eta(q^4)^4 * eta(q^6)) in powers of q.
%F Euler transform of period 12 sequence [ 6, -7, 4, -3, 6, -8, 6, -3, 4, -7, 6, -4, ...].
%F a(n) = A113421(6*n + 1) = A124815(6*n + 1).
%F a(2*n + 1) = 6 * A260518(n). - _Michael Somos_, Oct 07 2015
%e G.f. = 1 + 6*x + 14*x^2 + 18*x^3 + 21*x^4 + 30*x^5 + 38*x^6 + 42*x^7 + ...
%e G.f. = q + 6*q^7 + 14*q^13 + 18*q^19 + 21*q^25 + 30*q^31 + 38*q^37 + ...
%t a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 1}, DivisorSum[ m, # KroneckerSymbol[ -3, #] KroneckerSymbol[ -4, m/#] &]]];
%t a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 1}, DivisorSum[ m, m/# KroneckerSymbol[ 12, #] &]]];
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^4 EllipticTheta[ 4, 0, x^3] / EllipticTheta[ 4, 0, x], {x, 0, n}];
%o (PARI) {a(n) = my(m = 6*n + 1); if (n<0, 0, sumdiv( m, d, d * kronecker( -3, d) * kronecker( -4, m/d)))};
%o (PARI) {a(n) = my(m = 6*n + 1); if (n<0, 0, sumdiv( m, d, m/d * kronecker( 12, d)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^13 * eta(x^3 + A)^2 / (eta(x + A)^6 * eta(x^4 + A)^4 * eta(x^6 + A)), n))};
%Y Cf. A113421, A124815, A260518.
%K nonn
%O 0,2
%A _Michael Somos_, Jul 16 2015
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