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Search: id:A115995
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| A115995 |
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Sum of the sizes of the Durfee squares of all partitions of n. |
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+0
9
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| 0, 1, 2, 3, 6, 9, 16, 23, 36, 52, 76, 106, 152, 207, 286, 386, 522, 691, 920, 1202, 1576, 2038, 2636, 3373, 4320, 5478, 6944, 8738, 10984, 13717, 17116, 21232, 26308, 32441, 39944, 48977, 59970, 73147, 89090, 108151, 131090, 158417, 191166, 230049
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Also sum of positive cranks of all partitions of n, n>1; see A064391. - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 20 2006
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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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LINKS
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Eric Weisstein's World of Mathematics, Durfee Square.
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FORMULA
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G.f.=sum(kz^(k^2)/product((1-z^j)^2,j=1..k),k=1..infinity).
a(n) = sum_{k=1}^{floor(sqrt(n))} k*A115994(n,k).
Convolution of A067742 and A000041. - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 20 2006
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EXAMPLE
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a(4)=6 because the partitions [4],[3,1],[2,2],[2,1,1] and
[1,1,1,1] of 4 have Durfee squares of sizes 1,1,2,1 and 1, respectively.
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MAPLE
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g:=sum(k*z^(k^2)/product((1-z^j)^2, j=1..k), k=1..10): gser:=series(g, z=0, 56): seq(coeff(gser, z, n), n=1..52);
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CROSSREFS
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Cf. A115994, A115720, A115721, A115722.
Sequence in context: A067435 A035494 A003244 this_sequence A051057 A147364 A147227
Adjacent sequences: A115992 A115993 A115994 this_sequence A115996 A115997 A115998
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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 11 2006
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EXTENSIONS
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Edited and verified by Frank Adams-Watters (FrankTAW(AT)Netscape net) Mar 11 2006
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