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Search: id:A116365
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| A116365 |
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Sum of the sizes of 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, 20, 33, 56, 86, 136, 200, 301, 429, 621, 868, 1219, 1669, 2297, 3091, 4171, 5542, 7357, 9648, 12652, 16402, 21250, 27298, 35003, 44556, 56637, 71515, 90160, 113046, 141464, 176189, 219053, 271149, 335044, 412447, 506787, 620597
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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a(n)=Sum(k*A114087(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..infty)}]_{t=1}.
a(n) = (n*A000041(n)-A116503(n))/2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), 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 total size 0+1+0+2+3=6.
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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..48);
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CROSSREFS
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Cf. A115994, A115995, A114087, A114088, A114089.
Adjacent sequences: A116362 A116363 A116364 this_sequence A116366 A116367 A116368
Sequence in context: A116557 A001911 A020957 this_sequence A055417 A018918 A077855
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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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