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A281452 Expansion of f(x, x) * f(x^5, x^13) in powers of x where f(, ) is Ramanujan's general theta function.

%I A281452 #16 Feb 16 2025 08:33:39
%S A281452 1,2,0,0,2,1,2,0,0,4,0,0,0,1,4,0,2,2,0,0,0,2,2,0,0,2,0,0,1,4,2,0,2,0,
%T A281452 0,0,2,2,2,0,0,2,0,0,3,2,0,0,2,4,0,0,0,4,2,0,0,0,0,0,2,0,2,0,4,0,0,0,
%U A281452 0,5,2,0,0,2,0,0,0,4,2,0,2,2,0,0,0,2,2
%N A281452 Expansion of f(x, x) * f(x^5, x^13) in powers of x where f(, ) is Ramanujan's general theta function.
%H A281452 G. C. Greubel, <a href="/A281452/b281452.txt">Table of n, a(n) for n = 0..1000</a>
%H A281452 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A281452 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A281452 f(a,b) = 1 + Sum_{k=1..oo} (ab)^(k(k-1)/2)*(a^k+b^k). - _N. J. A. Sloane_, Jan 30 2017
%F A281452 Euler transform of a period 36 sequence.
%F A281452 G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 4*k)).
%F A281452 G.f.: Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 - x^(18*k-13)) * (1 - x^(18*k-5)) * (1 - x^(18*k)).
%F A281452 a(n) = A122865(3*n + 1) = A122856(6*n + 2) = A258278(6*n + 2). a(n) = - A256269(9^n + 4). 4 * a(n) = A004018(9*n + 4).
%F A281452 a(n) = b(9*n + 4) with b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
%F A281452 2 * a(n) = b(9*n + 4) = with b = A105673, A105673, A122857, A258034, A259761. -2 * a(n) = b(9*n + 4) with b = A138949, A256280, A258292.
%F A281452 a(4*n) = A281453(n). a(8*n + 6) = 2 * A281490(n). a(16*n + 12) = A281451(n).
%F A281452 a(32*n + 4) = 2 * A281492(n). a(64*n + 28) = A281452(n). a(128*n + 60) = 2 * A281491(n).
%F A281452 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - _Amiram Eldar_, Jan 20 2025
%e A281452 G.f. = 1 + 2*x + 2*x^4 + x^5 + 2*x^6 + 4*x^9 + x^13 + 4*x^14 + 2*x^16 + ...
%e A281452 G.f. = q^4 + 2*q^13 + 2*q^40 + q^49 + 2*q^58 + 4*q^85 + q^121 + 4*q^130 + ...
%t A281452 a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 4, KroneckerSymbol[ -4, #] &]];
%t A281452 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^5, x^18] QPochhammer[ -x^13, x^18] QPochhammer[ x^18], {x, 0, n}];
%t A281452 a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 4])];
%o A281452 (PARI) {a(n) = if( n<0, 0, sumdiv(9*n + 4, d, (d%4==1) - (d%4==3)))};
%o A281452 (PARI) {a(n) = if( n<0, 0, my(m = 9*n + 4, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 2 || k%9 == 7), s+=(j>0)+1)); s)};
%o A281452 (PARI) {a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 4); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%4==1, e+1, 1-e%2)))};
%Y A281452 Cf. A002654, A004018, A019670, A035154, A105673, A113446, A122856, A122857, A122864, A122865.
%Y A281452 Cf. A125061, A129448, A138949, A138950, A163746, A256269, A256276, A256280, A258034.
%Y A281452 Cf. A258228, A258256, A258278, A258292, A259761.
%Y A281452 Cf. A281451, A281453, A281490, A281491, A281492.
%K A281452 nonn
%O A281452 0,2
%A A281452 _Michael Somos_, Jan 26 2017