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A281640 Expansion of x * f(x, x) * f(x^5, x^25) in powers of x where f(, ) is Ramanujan's general theta function.

%I A281640 #12 Feb 16 2025 08:33:40
%S A281640 1,2,0,0,2,1,2,0,0,4,0,0,0,0,2,0,2,0,0,0,0,2,0,0,0,3,2,0,0,2,2,0,0,0,
%T A281640 2,0,2,0,0,0,1,6,0,0,2,0,0,0,0,4,2,0,0,0,2,0,2,0,0,0,0,2,0,0,2,2,0,0,
%U A281640 0,2,0,0,0,0,2,0,2,0,0,0,1,4,0,0,2,0,2
%N A281640 Expansion of x * f(x, x) * f(x^5, x^25) in powers of x where f(, ) is Ramanujan's general theta function.
%H A281640 G. C. Greubel, <a href="/A281640/b281640.txt">Table of n, a(n) for n = 1..1000</a>
%H A281640 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A281640 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A281640 G.f.: x * (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(15*k^2 - 10*k)).
%F A281640 G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(30*k-25)) * (1 + x^(30*k-5)) * (1 - x^(30*k)).
%F A281640 a(n) = A122855(3*n + 2) = A260649(3*n + 2) = A122856(6*n + 4) = A258276(6*n + 4).
%F A281640 a(n) = - A140727(3*n + 2). 2 * a(n) = A192323(3*n + 2).
%e A281640 G.f. = x + 2*x^2 + 2*x^5 + x^6 + 2*x^7 + 4*x^10 + 2*x^15 + 2*x^17 + 2*x^22 + ...
%e A281640 G.f. = q^5 + 2*q^8 + 2*q^17 + q^20 + 2*q^23 + 4*q^32 + 2*q^47 + 2*q^53 + 2*q^65 + ...
%t A281640 a[ n_] := If[ n < 1, 0, DivisorSum[ 3 n + 2, KroneckerSymbol[ -15, #] (-1)^Boole[Mod[#, 4] == 2] &]];
%t A281640 a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] QPochhammer[ -x^5, x^30] QPochhammer[ -x^25, x^30] QPochhammer[ x^30], {x, 0, n}];
%o A281640 (PARI) {a(n) = if( n<1, 0, sumdiv(3*n + 2, d, kronecker(-15, d) * (-1)^(d%4==2) ))};
%o A281640 (PARI) {a(n) = if( n<1, 0, my(m = 3*n + 2, s, x); for(y=1, sqrtint(m\5), if( y%3 && issquare((m - 5*y^2)\3, &x), s += (x>0) + 1)); s)};
%o A281640 (PARI) {a(n) = if( n<1, 0, my(A); n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^10 + A)^2 * eta(x^15 + A) * eta(x^60 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^5 + A) * eta(x^20 + A) * eta(x^30 + A)), n))};
%Y A281640 Cf. A122855, A122856, A140727, A192323, A258276, A260649.
%K A281640 nonn
%O A281640 1,2
%A A281640 _Michael Somos_, Jan 25 2017