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A281490 Expansion of f(x, x^3) * f(x, x^8) in powers of x where f(, ) is Ramanujan's general theta function.

%I A281490 #23 Feb 16 2025 08:33:39
%S A281490 1,2,1,1,1,0,1,1,1,1,1,3,1,0,2,1,1,1,1,0,0,2,1,1,0,1,2,0,2,2,1,2,1,1,
%T A281490 0,1,3,1,0,1,2,0,0,0,1,2,2,1,0,0,0,2,1,2,1,1,2,1,2,1,0,3,0,1,1,0,4,1,
%U A281490 1,0,1,0,1,1,1,1,0,1,1,3,1,0,0,0,0,1,3
%N A281490 Expansion of f(x, x^3) * f(x, x^8) in powers of x where f(, ) is Ramanujan's general theta function.
%H A281490 G. C. Greubel, <a href="/A281490/b281490.txt">Table of n, a(n) for n = 0..1000</a>
%H A281490 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A281490 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A281490 f(x,x^m) = 1 + Sum_{k=1..oo} x^((m+1)*k*(k-1)/2)*(x^k + x^(m*k)). - _N. J. A. Sloane_, Jan 30 2017
%F A281490 G.f.: (Sum_{k>0} x^(k*(k - 1)/2)) * (Sum_{k in Z} x^(k*(9*k + 7)/2)).
%F A281490 G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)) * (1 + x^(9*k-8)) * (1 + x^(9*k-1)) * (1 - x^(9*k)).
%F A281490 Convolution of sequences A010054 and A281814.
%F A281490 2 * a(n) = A281451(32*n + 25).
%e A281490 G.f. = 1 + 2*x + x^2 + x^3 + x^4 + x^6 + x^7 + x^8 + x^9 + x^10 + 3*x^11 + ...
%e A281490 G.f. = q^29 + 2*q^65 + q^101 + q^137 + q^173 + q^245 + q^281 + q^317 + q^353 + ...
%t A281490 a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ -x, x^9] QPochhammer[ -x^8, x^9] QPochhammer[ x^9], {x, 0, n}];
%o A281490 (PARI) {a(n) = if( n<0, 0, sumdiv(36*n + 29, d, kronecker(-4, d)) / 2)};
%o A281490 (PARI) {a(n) = if( n<0, 0, my(A, p, e); n = 36*n + 29; A = factor(n); prod(k=1, matsize(A) [1], [p, e] = A[k, ]; if(p%4==1, e+1, 1-e%2)) / 2)};
%Y A281490 Cf. A010054, A281451, A281814.
%K A281490 nonn
%O A281490 0,2
%A A281490 _Michael Somos_, Jan 29 2017