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Search: id:A045831
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| A045831 |
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Number of 4-core partitions of n. |
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4
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| 1, 1, 2, 3, 1, 3, 3, 3, 4, 4, 2, 2, 7, 3, 5, 6, 2, 4, 7, 3, 4, 7, 5, 8, 5, 4, 4, 8, 5, 6, 7, 2, 9, 11, 3, 8, 9, 4, 6, 5, 7, 5, 14, 7, 4, 10, 5, 10, 11, 3, 9, 10, 5, 8, 10, 4, 6, 15, 8, 9, 10, 6, 8, 15, 6, 10, 6, 5, 15, 9, 6, 8, 14, 8, 6, 13, 5, 16, 18, 7, 8, 7, 9, 6, 15, 6, 12, 17, 5, 8, 15, 7, 12
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Euler transform of period 4 sequence [1,1,1,-3,...].
Expansion of eta(q^4)^4/(eta(q)q^(5/8)) in powers of q.
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REFERENCES
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Hirschhorn, M. and Sellers, J., Some amazing facts about 4-cores, J. Num. Thy. 60 (1996), 51-69.
Ono, K. and Sze, L., 4-core partitions and class numbers, Acta. Arith. 80 (1997), 249-272.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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eta(32z)^4/eta(8z) = Sum q^(x^2+2y^2+2z^2), x, y, z >= 1 and odd.
Number of solutions to n=t1+2*t2+2*t3 where t1, t2, t3 are triangular numbers. - Michael Somos Jan 02 2006
G.f.: Product_{k>0} (1-q^(4k))^4/(1-q^k).
Expansion of psi(q) * psi(q^2)^2 in powers of q where psi() is a Ramanujan theta function. - Michael Somos Sep 02 2008
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EXAMPLE
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q^5 + q^13 + 2*q^21 + 3*q^29 + q^37 + 3*q^45 + 3*q^53 + 3*q^61 + 4*q^69 + ...
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PROGRAM
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(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)^4/eta(x+A), n))}
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CROSSREFS
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A004024/4.
Sequence in context: A104483 A080717 A121062 this_sequence A046821 A030306 A119348
Adjacent sequences: A045828 A045829 A045830 this_sequence A045832 A045833 A045834
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms and additional comments from James A. Sellers (sellersj(AT)math.psu.edu), Feb 11 2000
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