A124815 Expansion of q * psi(q)^2 * psi(-q^3)^2 * phi(-q^6) / phi(-q^2) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 12, 12, 14, 12, 12, 16, 16, 18, 18, 16, 18, 24, 24, 24, 21, 28, 27, 24, 28, 24, 30, 32, 36, 32, 24, 36, 38, 36, 42, 32, 40, 36, 42, 48, 36, 48, 48, 48, 43, 42, 48, 56, 52, 54, 48, 48, 54, 56, 60, 48, 62, 60, 54, 64, 56, 72, 66, 64, 72, 48, 72, 72

Offset: 1

Author

Michael Somos, Nov 08 2006

Keywords

• nonn
• easy
• mult

Comments

Number 24 of the 74 eta-quotients listed in Table I of Martin (1996).

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

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
• Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852, Table I.
• Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A258414.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ n/d KroneckerSymbol[ 12, d], { d, Divisors[ n]}]]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^2 QPochhammer[ q^3]^2 QPochhammer[ q^4] QPochhammer[ q^12]/QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

PARI

{a(n) = if( n<1, 0, sumdiv( n, d, n/d * kronecker( 12, d)))};

PARI

{a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; f = kronecker( 12, p); (p^(e+1) - f^(e+1)) / (p - f)))};

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^12 + A) / eta(x + A)^2, n))};

Formula

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

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

a(n) is multiplicative with a(p^e) = p^e if p<5, a(p^e) = (p^(e+1) - 1) / (p-1) if p == 1, 11 (mod 12), a(p^e) = (p^(e+1) + (-1)^e) / (p+1) if p == 5, 7 (mod 12).

G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 - x^(2*k) + x^(4*k)).

G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(4*k)) * (1 - x^(12*k)).

a(2*n) = 2 * a(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(6*sqrt(3)) = 0.949703... (A258414). - Amiram Eldar, Dec 22 2023