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Search: id:A114087
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| A114087 |
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Triangle read by rows: T(n,k) is number of partitions of n whose tails below their Durfee squares have size k (n>=1; 0<=k<=n-1). |
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+0
4
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| 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 2, 3, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 4, 3, 5, 3, 4, 1, 1, 1, 5, 4, 5, 5, 4, 4, 1, 1, 1, 6, 5, 7, 5, 7, 4, 5, 1, 1, 1, 7, 6, 9, 7, 7, 7, 5, 5, 1, 1, 1, 9, 7, 11, 10, 10, 7, 9, 5, 6, 1, 1, 1, 10, 9, 13, 12, 14, 10, 9, 9, 6, 6, 1, 1, 1, 12, 10, 17, 15, 17, 15
(list; table; graph; listen)
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OFFSET
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1,7
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COMMENT
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Row sums yield A000041. Column 0 is A003114. Sum(k*T(n,k),k=0..n-1)=A116365(n).
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
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FORMULA
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G.f.=sum(q^(k^2)/product((1-q^j)(1-(t*q)^j),j=1..k),k=1..infinity).
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EXAMPLE
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T(6,2)=3 because we have [4,1,1], [2,2,2], and [2,2,1,1] (the bottom tails are [1,1], [2], and [1,1], respectively, each being a partition of 2).
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MAPLE
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g:=sum(z^(k^2)/product((1-z^j), j=1..k)/product((1-(t*z)^i), i=1..k), k=1..20): gserz:=simplify(series(g, z=0, 30)): for n from 1 to 14 do P[n]:=coeff(gserz, z^n) od: for n from 1 to 14 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A115994, A115995, A116365, A114088, A114089.
Sequence in context: A058398 A091499 A137350 this_sequence A008284 A114088 A037306
Adjacent sequences: A114084 A114085 A114086 this_sequence A114088 A114089 A114090
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 12 2006
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