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Search: id:A114089
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| A114089 |
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Total number of parts in the tails below the Durfee squares of all partitions of n. |
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+0
4
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| 0, 1, 3, 6, 11, 19, 31, 50, 76, 116, 169, 247, 349, 494, 682, 941, 1274, 1724, 2296, 3054, 4014, 5263, 6833, 8854, 11373, 14578, 18556, 23561, 29736, 37447, 46903, 58619, 72925, 90518, 111899, 138044, 169665, 208111, 254436, 310456, 377687, 458625
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OFFSET
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1,3
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COMMENT
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a(n)=Sum(k*A114088(n,k),k=0..n-1).
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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.=[(d/dt){sum(q^(k^2)/product((1-q^j)(1-tq^j),j=1..k),k=1..infinity)}]_{t=1}.
a(n) = A006128(n) - A115995(n). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 18 2006
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EXAMPLE
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a(4)=6 because the bottom tails of the five partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], are { }, [1], { }, [1,1] and [1,1,1], respectively, having a total of 6 parts.
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MAPLE
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g:=sum(z^(k^2)/product((1-z^j)*(1-t*z^j), j=1..k), k=1..10): dgdt1:=simplify(subs(t=1, diff(g, t))): dgdt1ser:=series(dgdt1, z=0, 55): seq(coeff(dgdt1ser, z, n), n=1..45);
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CROSSREFS
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Cf. A115994, A115995, A114087, A116365, A114088.
Sequence in context: A116100 A004133 A050228 this_sequence A001976 A144115 A116557
Adjacent sequences: A114086 A114087 A114088 this_sequence A114090 A114091 A114092
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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 12 2006
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