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A053724 Number of 7-core partitions of n. 7
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 (list; graph; refs; listen; history; text; internal format)
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: A087174 A071963 A224075 * A046220 A141792 A180458

Adjacent sequences:  A053721 A053722 A053723 * A053725 A053726 A053727

KEYWORD

easy,nonn

AUTHOR

James A. Sellers, Feb 11 2000

STATUS

approved

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Last modified May 27 06:24 EDT 2019. Contains 323599 sequences. (Running on oeis4.)