A161528 Expansion of the q-series Sum_{n >= 0} (-1)^nq^(n(n+1)/2)(1-q)(1-q^2)...(1-q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).

1, 0, 2, 1, 0, 0, 2, 1, 2, 0, 0, 0, 2, 0, 2, 2, 1, 0, 0, 0, 2, 0, 0, 2, 3, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 2, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 4, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 2, 2, 1, 0, 4, 0, 2, 0, 0, 0, 0, 0, 2, 2

Offset: 0

Author

Jeremy Lovejoy, Jun 12 2009

Keywords

• nonn

Links

• Jeremy Lovejoy and Olivier Mallet, n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities, South East Asian J. Math. Math. Sci., Vol. 6, No. 2 (2008), pp. 23-36.

Crossrefs

Cf. A001622, A035187.

Programs

Mathematica

f[p_, e_] := If[MemberQ[{2, 3}, Mod[p, 5]], (1 + (-1)^e)/2, e+1]; f[5, e_] := 1; a[0] = 1; a[n_] := Times @@ f @@@ FactorInteger[5*n+1]; Array[a, 100, 0] (* Amiram Eldar, Jan 11 2025 *)

PARI

a(n) = {my(f = factor(5*n+1)); prod(i = 1, #f~, if(f[i, 1] == 5, 1, if(f[i, 1] % 5 == 2 || f[i, 1] % 5 == 3, (1 + (-1)^f[i, 2])/2, f[i, 2] + 1)));} \\ Amiram Eldar, Jan 11 2025

Formula

a(n) = A035187(5n+1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*log(phi)/sqrt(5) = 0.860817..., where phi is the golden ratio (A001622) . - Amiram Eldar, Jan 11 2025