A133657 Expansion of q * (phi(q) * psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

1, 4, 4, 0, 6, 16, 8, 0, 13, 24, 12, 0, 14, 32, 24, 0, 18, 52, 20, 0, 32, 48, 24, 0, 31, 56, 40, 0, 30, 96, 32, 0, 48, 72, 48, 0, 38, 80, 56, 0, 42, 128, 44, 0, 78, 96, 48, 0, 57, 124, 72, 0, 54, 160, 72, 0, 80, 120, 60, 0, 62, 128, 104, 0, 84, 192, 68, 0, 96

Offset: 1

Author

Michael Somos, Sep 20 2007

Keywords

• nonn
• mult

Comments

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

Examples

			G.f. = q + 4*q^2 + 4*q^3 + 6*q^5 + 16*q^6 + 8*q^7 + 13*q^9 + 24*q^10 + ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A008438, A097723, A112610, A121455, A133690, A222068.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^2]/2)^2, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
a[n_] := Switch[IntegerExponent[n, 2], 0, DivisorSigma[1, n], 1, 4*DivisorSigma[1, n/2], , 0]; Array[a, 100] (* _Amiram Eldar, Nov 12 2022 *)

PARI

{a(n) = if( n<1, 0, if( n%2, sigma(n), if( n%4, 4 * sigma(n/2), 0)))};

PARI

{a(n) = my(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3) )^2, n))};

Formula

Expansion of (eta(q^2)^5 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.

Euler transform of period 8 sequence [ 4, -6, 4, 0, 4, -6, 4, -4, ...].

a(n) is multiplicative with a(2) = 4, a(2^e) = 0 if e>1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133690.

a(4*n) = 0. a(4*n+2) = 4 * sigma(2*n+1). a(2*n+1) = sigma(2*n+1).

a(n) = -(-1)^n * A121455(n). Convolution square of A113411.

a(2*n + 1) = A008438. a(4*n + 1) = A112610(n). a(4*n + 3) = 4 * A097723(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/16 = 0.6168502... (A222068). - Amiram Eldar, Nov 12 2022