A122865 Expansion of chi(x) * phi(x^3) * psi(-x^3) in powers of x where chi(), phi(), psi() are Ramanujan theta functions.

1, 1, 0, 2, 2, 1, 0, 0, 3, 0, 0, 2, 2, 2, 0, 0, 1, 2, 0, 2, 2, 1, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 2, 3, 0, 2, 2, 0, 0, 0, 1, 0, 0, 4, 0, 2, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 2, 1, 0, 2, 4, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 4, 0, 1, 0

Offset: 0

Author

Michael Somos, Sep 15 2006

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + 2*x^3 + 2*x^4 + x^5 + 3*x^8 + 2*x^11 + 2*x^12 + 2*x^13 + ...
G.f. = q + q^4 + 2*q^10 + 2*q^13 + q^16 + 3*q^25 + 2*q^34 + 2*q^37 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A002175, A002654, A035154, A113446, A122856, A122864, A163746, A258277.

Programs

Mathematica

phi[q_] := EllipticTheta[3, 0, q]; chi[q_] := ((1 - InverseEllipticNomeQ[q]) * InverseEllipticNomeQ[q]/(16*q))^(-1/24); psi[q_] := (1/2)*q^(-1/8)*EllipticTheta[ 2, 0, q^(1/2)]; s = Series[ chi[q]*phi[q^3]*psi[-q^3], {q, 0, 104}]; a[n_] := Coefficient[s, q, n];
(* or *) a[n_] := If[n == 0, 1, Sum[Boole[Mod[d, 4] == 1] - Boole[Mod[d, 4] == 3], {d, Divisors[3n+1]}]]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Feb 17 2015, after PARI code *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 3 n + 1, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^4 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))};

PARI

{a(n) = my(A, p, e); if(n <0, 0, n = 3*n+1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p==3, -2*(-1)^e, p%4==1, e+1, 1-e%2)))};

PARI

{a(n) = if( n<0, 0, n = 3*n+1; sumdiv(n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Apr 19 2007 */

Formula

Expansion of chi(x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Sep 02 2015

Expansion of q^(-1/3) * eta(q^2)^2 * et(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [1, -1, 2, 0, 1, -4, 1, 0, 2, -1, 1, -2, ...]. - Michael Somos, Apr 19 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258228. - Michael Somos, Sep 02 2015

a(n) = A002654(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122864(3*n + 1) = A163746(3*n + 1).

a(n) = (-1)^n * A258277(n). a(2*n + 1) = A122856(n). - Michael Somos, Sep 02 2015

a(4*n) = A002175(n). a(4*n + 2) = 0. - Michael Somos, Jan 19 2017