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%I #33 May 19 2017 07:41:39
%S 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,
%T 82,133,112,120,120,165,122,180,147,186,176,225,168,255,210,245,224,
%U 324,219,338,276,341,294,385,288,441,352,410,366,518,360,506,435,504
%N Number of 7-core partitions of n.
%D 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.
%D 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).
%H Seiichi Manyama, <a href="/A053724/b053724.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A. Berkovich and H. Yesilyurt, <a href="http://dx.doi.org/10.1016/j.disc.2007.09.044">New identities for 7-cores with prescribed BG-rank</a>, Discrete Math., 308 (2008), 5246-5259.
%H F. Garvan, D. Kim and D. Stanton, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN00210752X&IDDOC=178508">Cranks and t-cores</a>, Inventiones Math. 101 (1990) 1-17,
%H B. Kim, <a href="http://dx.doi.org/10.1016/j.disc.2009.09.024">On inequalities and linear relations for 7-core partitions</a>, Discrete Math., 310 (2010), 861-868.
%H K. Saito, <a href="https://arxiv.org/abs/math/0602367">Eta-produkt eta(7tau)^7/eta(tau)</a>, arXiv:math/0602367 [math.NT], 2006.
%F Expansion of q^(-2) * eta(q^7)^7 / eta(q) in powers of q.
%F Euler transform of period 7 sequence [ 1, 1, 1, 1, 1, 1, -6, ...].
%F a(7*n + 5) == 0 (mod 7).
%F G.f.: Product_{k>0} (1 - q^(7*k))^7 / (1 - q^k).
%e 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 + ...
%e G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 8*q^9 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^7]^7 / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Feb 22 2015 *)
%o (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 */
%Y Cf. A053723, column t=7 of A175595.
%K easy,nonn
%O 0,3
%A _James A. Sellers_, Feb 11 2000
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