A053724 Number of 7-core partitions of n.

1, 1, 2, 3, 5, 7, 11, 8, 15, 16, 21, 21, 28, 24, 44, 36, 49, 45, 63, 49, 74, 64, 85, 72, 105, 82, 133, 112, 120, 120, 165, 122, 180, 147, 186, 176, 225, 168, 255, 210, 245, 224, 324, 219, 338, 276, 341, 294, 385, 288, 441, 352, 410, 366, 518, 360, 506, 435, 504

Offset: 0

Author

James Sellers, Feb 11 2000

Keywords

• easy
• nonn

Examples

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 8*x^7 + 15*x^8 + ...
G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 8*q^9 + ...
		

References

• A. Balog, H. Darmon, K. Ono, Congruence for Fourier coefficients of half-integral weight modular forms and special values of L-functions, pp. 105-128 of Analytic number theory, Vol. 1, Birkhauser, Boston, 1996, see page 107.
• B. Berndt, Commentary on Ramanujan's Papers, pp. 357-426 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 372 (4).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
• A. Berkovich and H. Yesilyurt, New identities for 7-cores with prescribed BG-rank, Discrete Math., 308 (2008), 5246-5259.
• F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17,
• B. Kim, On inequalities and linear relations for 7-core partitions, Discrete Math., 310 (2010), 861-868.
• K. Saito, Eta-produkt eta(7tau)^7/eta(tau), arXiv:math/0602367 [math.NT], 2006.

Crossrefs

Cf. A053723, column t=7 of A175595.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^7]^7 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^7 + A)^7 / eta(x + A), n))}; /* Michael Somos, Apr 16 2005 */

Formula

Expansion of q^(-2) * eta(q^7)^7 / eta(q) in powers of q.

Euler transform of period 7 sequence [ 1, 1, 1, 1, 1, 1, -6, ...].

a(7*n + 5) == 0 (mod 7).

G.f.: Product_{k>0} (1 - q^(7*k))^7 / (1 - q^k).