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Search: id:A116932
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| A116932 |
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Partitions of n in which each part, with the possible exception of the largest, occurs at least three times. |
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2
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| 1, 2, 2, 3, 3, 6, 6, 9, 12, 14, 16, 24, 25, 32, 40, 49, 56, 73, 81, 102, 120, 142, 162, 202, 227, 270, 316, 367, 419, 506, 565, 663, 767, 879, 998, 1179, 1317, 1517, 1739, 1979, 2232, 2588, 2883, 3295, 3742, 4220, 4737, 5426, 6037, 6828, 7701, 8642, 9651, 10939
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OFFSET
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1,2
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COMMENT
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Also, partitions of n in which any two distinct parts differ by at least 3. Example: a(5)=3 because we have [5],[4,1], and [1,1,1,1,1].
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FORMULA
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G.f.=sum(x^k*product(1+x^(3j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). More generally, the g.f. of partitions of n in which each part, with the possible exception of the largest, occurs at least b times, is sum(x^k*product(1+x^(bj)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). It is also the g.f. of partitions of n in which any two distinct parts differ by at least b.
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EXAMPLE
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a(5)=3 because we have [5],[2,1,1,1], and [1,1,1,1,1].
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MAPLE
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g:=sum(x^k*product(1+x^(3*j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..70): gser:=series(g, x=0, 62): seq(coeff(gser, x^n), n=1..58);
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CROSSREFS
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Cf. A116931.
Sequence in context: A133392 A101199 A032155 this_sequence A116450 A054172 A121211
Adjacent sequences: A116929 A116930 A116931 this_sequence A116933 A116934 A116935
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2006
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