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

0,3

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.

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).

Example

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 + ...

Mathematica

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

Prog

(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 */

Crossrefs

Cf. A053723, column t=7 of A175595.

Sequence in context: A071963 A224075 A354790 * A046220 A141792 A180458

Adjacent sequences: A053721 A053722 A053723 * A053725 A053726 A053727

Keyword

easy,nonn

Author

James A. Sellers, Feb 11 2000

Status

approved