A213627 Expansion of psi(x)^4 / psi(x^3) in powers of x where psi() is a Ramanujan theta function.

1, 4, 6, 7, 9, 6, 7, 15, 12, 12, 13, 6, 12, 18, 18, 13, 15, 18, 12, 24, 12, 13, 27, 12, 24, 15, 12, 24, 28, 30, 12, 27, 18, 12, 30, 18, 19, 27, 24, 24, 27, 24, 36, 30, 18, 19, 24, 24, 24, 45, 18, 12, 45, 30, 24, 28, 18, 36, 36, 36, 24, 15, 36, 36, 51, 18, 25

Offset: 0

Author

Michael Somos, Jun 16 2012

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 4*x + 6*x^2 + 7*x^3 + 9*x^4 + 6*x^5 + 7*x^6 + 15*x^7 + 12*x^8 + ...
G.f. = q + 4*q^9 + 6*q^17 + 7*q^25 + 9*q^33 + 6*q^41 + 7*q^49 + 15*q^57 + 12*q^65 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2501 terms from G. C. Greubel)
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A212907.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      add([-3, 4, -4, 3, -4, 4][1+irem(d, 6)]*d,
        d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 18 2020

Mathematica

a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^4 / EllipticTheta[ 2, 0, q^3], {q, 0, 2 n + 1/4}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A) / (eta(x + A)^4 * eta(x^6 + A)^2), n))};

Formula

Expansion of q^(-1/8) * eta(q^2)^8 * eta(q^3) / (eta(q)^4 * eta(q^6)^2) in powers of q.

a(3*n + 2) = 6 * A212907(n).

Euler transform of period 6 sequence [4, -4, 3, -4, 4, -3, ...]. - Georg Fischer, Aug 18 2020