A122857 Expansion of (phi(q)^2 + phi(q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

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

Offset: 0

Author

Michael Somos, Sep 14 2006

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*q + 2*q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 2*q^6 + 2*q^8 + 2*q^9 + 4*q^10 + ...
		

References

• B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 197, Entry 44.

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. A002654, A008441, A019693, A035154, A113446, A125079, A125061, A132003.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Jul 09 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 2, {q, 0, n}]; (* Michael Somos, Jul 09 2013 *)

PARI

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -36, d)))};

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<5, 1, p%12<6, e+1, !(e%2) )))};

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==3, 1+e%2*2, p%4==1, e+1, !(e%2) )))};

Formula

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

Expansion of 2 * psi(q) * psi(q^2) * psi(q^3) / psi(q^6) - phi(q^3)^2 in powers of q. - Michael Somos, Jul 09 2013

Euler transform of period 12 sequence [ 2, -1, 0, 0, 2, -4, 2, 0, 0, -1, 2, -2, ...].

Moebius transform is period 12 sequence [ 2, 0, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, ...].

a(12*n + 7) = a(12*n + 11) = 0.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A125061.

A035154(n) = a(n) / 2 if n > 0. A008441(n) = a(4*n + 1) / 2. A125079(n) = a(2*n + 1) / 2. A113446(3*n + 1) = A002654(3*n + 1) = a(3*n + 1) / 2.

a(n) = (-1)^n * A132003(n). Expansion of (phi(-q^3) / phi(-q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 05 2023

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.0943951... (A019693). - Amiram Eldar, Nov 21 2023