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Search: id:A098884
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| A098884 |
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Number of partitions of n into distinct parts in which each part is congruent to 1 or 5 mod 6. |
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+0
3
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| 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 1, 2, 3, 3, 2, 1, 1, 3, 5, 5, 3, 1, 2, 5, 7, 7, 5, 3, 3, 7, 11, 11, 7, 4, 6, 11, 15, 15, 11, 7, 8, 15, 22, 22, 15, 10, 13, 22, 30, 30, 23, 16, 18, 30, 42, 42, 31, 22, 27, 43, 56, 56, 44, 33, 37, 57, 77, 77, 59, 45, 53, 79, 101, 101, 82, 64, 71
(list; graph; listen)
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OFFSET
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0,13
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COMMENT
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G.f. is reciprocal of Prod_{k>0}(1+x^(2*k-1)+x^(4*k-2)), which gives the number of partitions of n into odd parts none appearing more than twice.
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REFERENCES
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Noureddine Chair, Partition Identities From Partial Supersymmetry hep-th/0409011 2004
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FORMULA
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G.f.:=1/product_{k>0}(1-x^(2*k-1)+x^(4*k-2))= product_{k>0}(1+x^(6*k-1))*(1+x^(6*k-5))
Euler transform of period 12 sequence [1, -1, 0, 0, 1, 0, 1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Jun 26 2005
Expansion of q^(-1/12) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)^2) in powers of q.
Expansion of G(x^6) * H(-x) + x * G(-x) * H(x^6) where G(), H() are Rogers-Ramanujan functions.
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EXAMPLE
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E.g. a(25)=5 because 25=19+5+1=17+7+1=13+7+5=13+11+1.
q + q^13 + q^61 + q^73 + q^85 + q^97 + q^133 + 2*q^145 + 2*q^157 + q^169 + ...
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MAPLE
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series(product((1+x^x^(6*k-1))*(1+x^(6*k-5)), k=1..100), x=0, 100);
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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^2+A)^2*eta(x^3+A)*eta(x^12+A)/eta(x+A)/ eta(x^4+A)/eta(x^6+A)^2, n))} /* Michael Somos Jun 26 2005 */
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CROSSREFS
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Cf. A003105.
Sequence in context: A061926 A053188 A109389 this_sequence A039800 A109246 A037890
Adjacent sequences: A098881 A098882 A098883 this_sequence A098885 A098886 A098887
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KEYWORD
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nonn
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AUTHOR
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Noureddine Chair (n.chair(AT)rocketmail.com), Oct 14 2004
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