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

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, 10, 24, 0, -16, 24, 16, -24, -20, 16, 16, 0, -10, 8, 24, -32, -8, 32, 16, 0, -24, 24, 16, -40, -24, 18, 28, 0, -8, 24, 32, -32, -16, 16, 8, 0, -32, 24, 32, -40, -26, 32

Offset

0,2

Comments

a(8n+3) = 0.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

K. Bringmann and J. Lovejoy, Overpartitions and class numbers of binary quadratic forms

Formula

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).

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

a(4*n) = A256624(n) = 2 * a(n) - A005875(n). - Michael Somos, Jul 13 2015

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

Example

G.f. = 1 + 2*x + 4*x^2 - 2*x^4 + 8*x^5 + 8*x^6 - 8*x^7 - 4*x^8 + ...

Mathematica

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 *)

Prog

(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 */
(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 */

Crossrefs

Cf. A005875, A045831, A213022, A213624, A213625, A256624.

Sequence in context: A281499 A256014 A256280 * A111818 A007631 A281765

Adjacent sequences: A153179 A153180 A153181 * A153183 A153184 A153185

Keyword

sign

Author

Jeremy Lovejoy, Dec 20 2008

Status

approved