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Search: id:A066183
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| A066183 |
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Total sum of squares of parts in all partitions of n. |
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+0
5
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| 1, 6, 17, 44, 87, 180, 311, 558, 910, 1494, 2302, 3608, 5343, 7986, 11554, 16714, 23549, 33270, 45942, 63506, 86338, 117156, 156899, 209926, 277520, 366260, 479012, 624956, 808935, 1044994, 1340364, 1715572, 2182935, 2770942, 3499379
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Sum of hook lengths of all boxes in the Ferrers diagrams of all partitions of n (see the Guo-Niu Han paper, p. 25, Corollary 6.5). Example: a(3)=17 because for the partitions (3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively; the total sum of all hook lengths is 6+5+6=17. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 15 2008
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REFERENCES
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Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008.
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FORMULA
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Sum_{k=1..n} sigma_2(k)*numbpart(n-k), where sigma_2(k)=sum of squares of divisors of k=A001157(k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 26 2002
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EXAMPLE
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a(3)=17 because the squares of all partitions of 3 are {9},{4,1} and {1,1,1}, summing to 17.
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MATHEMATICA
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Table[Apply[Plus, Partitions[n]^2, {0, 2}], {n, 30}]
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CROSSREFS
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Cf. A000041, A001157.
Sequence in context: A047861 A099858 A062020 this_sequence A048746 A026382 A054492
Adjacent sequences: A066180 A066181 A066182 this_sequence A066184 A066185 A066186
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KEYWORD
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easy,nonn
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AUTHOR
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Wouter Meeussen (wouter.meeussen(AT)pandora.be), Dec 15 2001
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EXTENSIONS
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More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), Feb 07 2002
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