A258256 Expansion of f(q^3) * psi(-q^3)^3 / (psi(-q) * psi(-q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

1, 1, 1, 0, 1, 2, 0, 0, 1, 4, 2, 0, 0, 2, 0, 0, 1, 2, 4, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 4, 2, 0, 0, 2, 2, 0, 0, 0, 8, 0, 0, 0, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 4, 2, 2, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 4, 0

Offset: 0

Author

Michael Somos, May 24 2015

Keywords

• nonn
• easy
• mult

Comments

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

Examples

			G.f. = 1 + q + q^2 + q^4 + 2*q^5 + q^8 + 4*q^9 + 2*q^10 + 2*q^13 + q^16 + ...
		

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. A002175, A019670, A089801, A121444, A122856, A122856, A256282, A258260.

Cf. A000122, A000700, A010054, A121373.

Programs

Magma

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

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, {1, 2, -1, 0}[[Mod[#, 4, 1]]] If[ Divisible[ #, 9], 4, 1] (-1)^(Boole[Mod[#, 8] == 6] + n + #) &]];
a[ n_] := If[ n < 2, Boole[n >= 0], Times @@ (Which[ # == 2, 1, Mod[#, 4] == 1, #2 + 1, True, If[# == 3, 4, 1] Mod[#2 + 1, 2]] & @@@ FactorInteger[n])];
a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]^3 / (Sqrt[2] EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)]), {q, 0, n}];

PARI

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

PARI

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

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};

Formula

Expansion of eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18) / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36)) in powers of q.

Euler transform of period 36 sequence [1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, ...].

Moebius transform is period 36 sequence [1, 0, -1, 0, 1, 0, -1, 0, 4, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -4, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).

a(2*n) = a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 3) = 0. a(12*n + 1) = A002175. a(12*n + 5) = 2 * A121444(n).

a(n) = Sum_{d|n} A258260(d) * (-1)^(n+d) if n>0.

a(n) = (-1)^n * A256282(n). - Michael Somos, Jun 06 2015

a(n) is multiplicative with a(0) = 1, a(2^e) = 1, a(3^e) = 2*(1 + (-1)^e), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jun 06 2015

Expansion of A0(x)^2 + A0(x)*A1(x) + A1(x)^2 in powers of x where A0(x) = phi(x^9), A1(x) = x * f(x^3, x^15) = x * A089801(x^3). - Michael Somos, Jun 23 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - Amiram Eldar, Nov 24 2023