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

0,2

Comments

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

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

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

Example

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 + ...

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}];

Prog

(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))};

Crossrefs

Cf. A212907.

Sequence in context: A024555 A363375 A269330 * A225871 A288383 A001690

Adjacent sequences: A213624 A213625 A213626 * A213628 A213629 A213630

Keyword

nonn

Author

Michael Somos, Jun 16 2012

Status

approved