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A053724
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Number of 7-core partitions of n.
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7
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| Euler transform of period 7 sequence [1,1,1,1,1,1,-6,...].
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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.
A. Berkovich and H. Yesilyurt, New identities for 7-cores ..., Discrete Math., 308 (2008), 5246-5259.
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).
Garvan, F., Kim, D. and Stanton, D., 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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
K. Saito, Eta-produkt eta(7tau)^7/eta(tau)
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FORMULA
| Expansion of q^(-2)eta(q^7)^7/eta(q) in powers of q.
a(7n+5)==0 mod 7.
G.f.: Product_{k>0} (1-q^(7k))^7/(1-q^k).
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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 */
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CROSSREFS
| Cf. A053723, column t=7 of A175595.
Sequence in context: A111679 A087174 A071963 * A046220 A141792 A180458
Adjacent sequences: A053721 A053722 A053723 * A053725 A053726 A053727
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KEYWORD
| easy,nonn
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AUTHOR
| James A. Sellers (sellersj(AT)math.psu.edu), Feb 11 2000
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