A259761 Expansion of (phi(x)^2 + phi(x^9)^2) / 2 in powers of x where phi() is a Ramanujan theta function.

1, 2, 2, 0, 2, 4, 0, 0, 2, 4, 4, 0, 0, 4, 0, 0, 2, 4, 4, 0, 4, 0, 0, 0, 0, 6, 4, 0, 0, 4, 0, 0, 2, 0, 4, 0, 4, 4, 0, 0, 4, 4, 0, 0, 0, 8, 0, 0, 0, 2, 6, 0, 4, 4, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 2, 8, 0, 0, 4, 0, 0, 0, 4, 4, 4, 0, 0, 0, 0, 0, 4, 4, 4, 0, 0, 8, 0

Offset: 0

Author

Michael Somos, Jul 04 2015

Keywords

• nonn
• easy

Comments

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

Examples

			G.f. = 1 + 2*x + 2*x^2 + 2*x^4 + 4*x^5 + 2*x^8 + 4*x^9 + 4*x^10 + 4*x^13 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A004018, A019685, A122856.

Cf. A000122, A000700, A010054, A121373.

Programs

Magma

A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[3] + 2*A[5] + 4*A[6] + 2*A[9] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[18] + 4*A[19];

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^2 + EllipticTheta[ 3, 0, x^9]^2) / 2, {x, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ ( Which[ # < 3, 1, # == 3, 1 + (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, (1 + (-1)^#2) / 2 ] & @@@ FactorInteger[n])];

PARI

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))};

Formula

phi(x) = 1 + 2*Sum_{m=1..oo} x^(m^2). - N. J. A. Sloane, Jan 30 2017

Expansion of phi(x) * phi(x^9) + 2 * x^2 * chi(x^3)^2 * psi(-x^9)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

a(n) = 2 * b(n) with a(0) = 1 and b() is multiplicative with b(2^e) = 1, b(3^e) = 1 + (-1)^e if e>0, b(p^e) = e+1 if p == 1, 5 (mod 12), (p^e) = (1 + (-1)^e)/2 if p == 7, 11 (mod 12).

a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.

a(2*n) = a(n). a(9*n) = A004018(n). a(6*n + 4) = 2 * A122856(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5*Pi/9 = 1.745329... (= 100 * A019685). - Amiram Eldar, Dec 29 2023