A121444 Expansion of f(x^3, x^9) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta functions.
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, 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, 1, 1, 3, 1, 0, 1, 0, 2, 0, 1, 1, 1, 2, 1, 0, 0, 1, 3, 2
Offset: 0
Examples
G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + ... G.f. = q^5 + q^17 + q^29 + q^41 + q^53 + 2*q^65 + q^89 + q^101 + q^113 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
- K. Saito, "Extended Affine Root Systems. V. Elliptic Eta-Products and Their Dirichlet Series", 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.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := If[ n < 0, 0, Sum[ I^d, {d, Divisors[12 n + 5]}] / (2 I)]; (* Michael Somos, Jul 25 2015 *) 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 *) 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 *) a[ n_] := If[ n < 0, 0, DivisorSum[ 12 n + 5, KroneckerSymbol[ -4, #] &] / 2]; (* Michael Somos, Nov 11 2015 *) 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 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *) -
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))}; -
PARI
{a(n) = if( n<0, 0, n = 12*n + 5; sumdiv(n, d, (d%4==1) - (d%4==3)) / 2)};
Formula
Expansion of f(-x^3) * f(-x^6) / chi(-x) in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-5/12) * eta(q^2) * eta(q^3) * eta(q^6) / eta(q) in powers of q.
Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, -2, ...].
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.
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)).
-2 * a(n) = A121363(3*n + 1).
Convolution square is A098098.
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
From Peter Bala, Jan 07 2021: (Start)
G.f. A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(12*n + 5)). See Agarwal, p. 285, equation 6.19.
A(x^2) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(12*n + 5)). Cf. A033761. (End)
Comments