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A121444 Expansion of f(x^3, x^9) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta functions.

%I A121444 #32 Feb 16 2025 08:33:02
%S A121444 1,1,1,1,1,2,0,1,1,1,2,1,1,0,1,2,1,0,2,1,1,1,1,1,1,2,1,0,0,1,2,2,1,1,
%T A121444 0,3,0,1,1,0,2,0,1,1,2,2,1,1,0,1,1,1,2,1,1,0,1,2,1,0,3,0,0,1,1,2,1,1,
%U A121444 1,1,3,1,0,1,0,2,0,1,1,1,2,1,0,0,1,3,2
%N A121444 Expansion of f(x^3, x^9) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta functions.
%C A121444 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A121444 Seiichi Manyama, <a href="/A121444/b121444.txt">Table of n, a(n) for n = 0..10000</a>
%H A121444 R. P. Agarwal, <a href="https://www.ias.ac.in/article/fulltext/pmsc/103/03/0269-0293">Lambert series and Ramanujan</a>, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
%H A121444 K. Saito, <a href="http://www.ams.org/mathscinet-getitem?mr=1881609">"Extended Affine Root Systems. V. Elliptic Eta-Products and Their Dirichlet Series"</a>, Proceedings on Moonshine and related topics (Montreal, QC, 1999), 139-161, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001.  MR1881609 (2003d:11066)  See page 215.
%H A121444 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A121444 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A121444 Expansion of f(-x^3) * f(-x^6) / chi(-x) in powers of x where chi(), f() are Ramanujan theta functions.
%F A121444 Expansion of q^(-5/12) * eta(q^2) * eta(q^3) * eta(q^6) / eta(q) in powers of q.
%F A121444 Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, -2, ...].
%F A121444 G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258210.
%F A121444 G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)).
%F A121444 -2 * a(n) = A121363(3*n + 1).
%F A121444 Convolution square is A098098.
%F A121444 a(n) = (-1)^n * A258832(n) = A052343(3*n + 1). -a(n) = A258291(3*n + 1). 2 * a(n) = A008441(3*n + 1). - _Michael Somos_, Jul 02 2015
%F A121444 From _Peter Bala_, Jan 07 2021: (Start)
%F A121444 G.f. A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(12*n + 5)). See Agarwal, p. 285, equation 6.19.
%F A121444 A(x^2) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(12*n + 5)). Cf. A033761. (End)
%e A121444 G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + ...
%e A121444 G.f. = q^5 + q^17 + q^29 + q^41 + q^53 + 2*q^65 + q^89 + q^101 + q^113 + ...
%t A121444 a[ n_] := If[ n < 0, 0, Sum[ I^d, {d, Divisors[12 n + 5]}] / (2 I)]; (* _Michael Somos_, Jul 25 2015 *)
%t A121444 a[ n_] := SeriesCoefficient[ 2 x^(3/8) QPochhammer[ x^6]^3 / (QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)]), {x, 0, n}]; (* _Michael Somos_, Jan 31 2015 *)
%t A121444 a[ n_] := Length @ FindInstance[ 24 n + 10 == (6 j + 3)^2 + (6 k + 1)^2 && j >= 0, {j, k}, Integers, 10^9]; (* _Michael Somos_, Jul 02 2015 *)
%t A121444 a[ n_] := If[ n < 0, 0, DivisorSum[ 12 n + 5, KroneckerSymbol[ -4, #] &] / 2]; (* _Michael Somos_, Nov 11 2015 *)
%t A121444 a[ n_] := If[ n < 0, 0, Sum[ Boole[ Mod[d, 4] == 1] - Boole[ Mod[d, 4] == 3], {d, Divisors[12 n + 5]}] / 2]; (* _Michael Somos_, Nov 11 2015 *)
%t A121444 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^3] QPochhammer[ x^6], {x, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)
%o A121444 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A), n))};
%o A121444 (PARI) {a(n) = if( n<0, 0, n = 12*n + 5; sumdiv(n, d, (d%4==1) - (d%4==3)) / 2)};
%Y A121444 Cf. A098098, A121363, A258210, A258291, A258832.
%K A121444 nonn,easy
%O A121444 0,6
%A A121444 _Michael Somos_, Jul 30 2006