cp's OEIS Frontend

A153182 Number of overpartitions of n with even M2-rank minus the number of overpartitions of n with odd M2-rank.

%I A153182 #12 Nov 29 2017 23:03:21
%S A153182 1,2,4,0,-2,8,8,-8,-4,10,8,0,-8,8,16,-16,-10,16,12,0,-8,16,8,-24,-8,
%T A153182 10,24,0,-16,24,16,-24,-20,16,16,0,-10,8,24,-32,-8,32,16,0,-24,24,16,
%U A153182 -40,-24,18,28,0,-8,24,32,-32,-16,16,8,0,-32,24,32,-40,-26,32
%N A153182 Number of overpartitions of n with even M2-rank minus the number of overpartitions of n with odd M2-rank.
%C A153182 a(8n+3) = 0.
%H A153182 G. C. Greubel, <a href="/A153182/b153182.txt">Table of n, a(n) for n = 0..1000</a>
%H A153182 K. Bringmann and J. Lovejoy, <a href="http://arxiv.org/abs/0712.0631">Overpartitions and class numbers of binary quadratic forms</a>
%F A153182 G.f.: 1 + 2Sum_{n >= 1} q^n(1+q)(1+q^3)...(1+q^(2n-1))/((1+q^2)(1+q^4)...(1+q^(2n))^2).
%F A153182 G.f.: (1 + 8 * Sum_{k>0} (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2) / (1 + 2 * Sum_{k>0} (-1)^k * x^k^2). - _Michael Somos_, Jul 13 2015
%F A153182 a(4*n) = A256624(n) = 2 * a(n) - A005875(n). - _Michael Somos_, Jul 13 2015
%F A153182 a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = 4 * A213625(n). a(8*n + 5) = 8 * A045831(n). a(8*n + 6) = 8 * A213624(n). - _Michael Somos_, Jul 13 2015
%e A153182 G.f. = 1 + 2*x + 4*x^2 - 2*x^4 + 8*x^5 + 8*x^6 - 8*x^7 - 4*x^8 + ...
%t A153182 a[n_]:= If[n < 0, 0, SeriesCoefficient[(1 + 8*Sum[(-1)^k x^(k^2 + 2*k)/(1 + x^(2*k))^2, {k, (Sqrt[4 n + 1] - 1)/2}])/EllipticTheta[4, 0, x], {x, 0, n}]]; Table[a[n], {n,0,30}] (* _G. C. Greubel_, Nov 29 2017 *)
%o A153182 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 + 8 * sum(k=1, sqrtint(n+1)-1, (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2, A)) / (1 + 2 * sum(k=1, sqrtint(n), (-1)^k * x^k^2, A)), n))}; /* _Michael Somos_, Jul 13 2015 */
%o A153182 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, n, x^k / (1 + x^(2*k)) * prod(j=1, k,  (1 + x^(2*j - 1)) / (1 + x^(2*j)), 1 + x * O(x^(n-k)))), n))}; /* _Michael Somos_, Jul 13 2015 */
%Y A153182 Cf. A005875, A045831, A213022, A213624, A213625, A256624.
%K A153182 sign
%O A153182 0,2
%A A153182 _Jeremy Lovejoy_, Dec 20 2008