cp's OEIS Frontend

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))).

%I A161528 #9 Jan 11 2025 03:10:25
%S A161528 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,
%T A161528 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,
%U A161528 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
%N 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))).
%H A161528 Jeremy Lovejoy and Olivier Mallet, <a href="https://rsmams.org/journals/articleinfo.php?articleid=298">n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities</a>, South East Asian J. Math. Math. Sci., Vol. 6, No. 2 (2008), pp. 23-36.
%F A161528 a(n) = A035187(5n+1).
%F A161528 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
%t A161528 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 *)
%o A161528 (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
%Y A161528 Cf. A001622, A035187.
%K A161528 nonn
%O A161528 0,3
%A A161528 _Jeremy Lovejoy_, Jun 12 2009