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%I #17 Mar 12 2021 22:24:47
%S 1,2,0,1,4,1,2,2,0,4,0,0,2,0,3,2,2,0,3,4,0,2,2,2,0,2,0,2,2,0,4,0,0,1,
%T 4,0,2,2,0,5,2,2,2,4,0,0,0,0,2,4,2,0,2,0,4,2,0,0,2,0,1,2,0,2,6,0,0,4,
%U 1,8,0,0,2,0,0,2,2,2,2,0,0,2,4,0,4,2,2
%N Expansion of phi(x) * f(x^3, x^5) 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="/A246862/b246862.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 Euler transform of period 16 sequence [ 2, -3, 3, -1, 3, -4, 2, -2, 2, -4, 3, -1, 3, -3, 2, -2, ...].
%F Convolution of A000122 and A214264.
%F a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = A246863(n).
%F a(n) = A113407(2*n) = A226192(2*n) = A008441(4*n) = A134343(4*n) = A116604(8*n) = A125079(8*n) = A129447(8*n) = A138741(8*n).
%e G.f. = 1 + 2*x + x^3 + 4*x^4 + x^5 + 2*x^6 + 2*x^7 + 4*x^9 + 2*x^12 + ...
%e G.f. = q + 2*q^17 + q^49 + 4*q^65 + q^81 + 2*q^97 + 2*q^113 + 4*q^145 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8], {x, 0, n}];
%o (PARI) {a(n) = if( n<0, 0, issquare(16 * n + 1) + 2 * sum(i=1, sqrtint(n), issquare(16 * (n - i^2) + 1)))};
%Y Cf. A000122, A008441, A113407, A116604, A125079, A129447, A134343, A138741, A214264, A226192, A246863.
%K nonn
%O 0,2
%A _Michael Somos_, Sep 05 2014
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