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A281491 Expansion of f(x, x^3) * f(x^2, x^7) in powers of x where f(, ) is Ramanujan's general theta function.

%I A281491 #13 Feb 16 2025 08:33:39
%S A281491 1,1,1,2,0,1,1,1,2,0,2,0,1,2,1,1,1,2,0,1,0,1,1,3,1,0,1,0,3,1,1,0,0,2,
%T A281491 2,1,2,0,2,1,0,1,0,2,1,1,0,1,2,2,0,2,1,0,2,1,0,1,2,1,0,1,2,2,0,1,1,0,
%U A281491 4,2,0,0,1,1,0,0,1,1,3,1,1,0,1,2,1,2,0
%N A281491 Expansion of f(x, x^3) * f(x^2, x^7) in powers of x where f(, ) is Ramanujan's general theta function.
%H A281491 G. C. Greubel, <a href="/A281491/b281491.txt">Table of n, a(n) for n = 0..1000</a>
%H A281491 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A281491 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A281491 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 A281491 Euler transform of period 18 sequence [1, 0, 1, -2, 1, -1, 2, -1, 0, -1, 2, -1, 1, -2, 1, 0, 1, -2, ...].
%F A281491 G.f.: (Sum_{k>0} x^(k*(k - 1)/2)) * (Sum_{k in Z} x^(k*(9*k + 5)/2)).
%F A281491 G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)) * (1 + x^(9*k-7)) * (1 + x^(9*k-2)) * (1 - x^(9*k)).
%F A281491 2 * a(n) = A281451(8*n + 3).
%e A281491 G.f. = 1 + x + x^2 + 2*x^3 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^10 + x^12 + ...
%e A281491 G.f. = q^17 + q^53 + q^89 + 2*q^125 + q^197 + q^233 + q^269 + 2*q^305 + ...
%t A281491 a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ -x^2, x^9] QPochhammer[ -x^7, x^9] QPochhammer[ x^9], {x, 0, n}];
%o A281491 (PARI) {a(n) = if( n<0, 0, sumdiv(36*n + 17, d, kronecker(-4, d)) / 2)};
%Y A281491 Cf. A281451.
%K A281491 nonn
%O A281491 0,4
%A A281491 _Michael Somos_, Jan 29 2017