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Search: id:A116465
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| A116465 |
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Sum of the sizes of the Durfee squares of all partitions of n into odd parts. |
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+0
2
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| 1, 1, 2, 2, 3, 5, 6, 8, 12, 16, 20, 27, 34, 43, 56, 68, 84, 104, 126, 153, 187, 222, 266, 317, 378, 445, 528, 620, 728, 853, 997, 1159, 1353, 1566, 1818, 2102, 2427, 2793, 3218, 3692, 4236, 4849, 5545, 6325, 7220, 8210, 9337, 10599, 12023, 13609, 15403, 17394
(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*A116464(n,k), k>=1).
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FORMULA
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G.f.=sum(2ix^(4i^2+2i)/[product(1-x^(2j),j=1..2i)product(1-x^(2j-1),j=1..i)], i=1..infinity)+ sum((2i-1)x^((2i-1)^2)/[product(1-x^(2j),j=1..2i-1)product(1-x^(2j-1),j=1..i)],i=1..infinity).
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EXAMPLE
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a(7)=6 because the partitions of 5 into odd parts are [7], [5,1,1], [3,3,1],
[3,1,1,1,1] and [1,1,1,1,1,1,1], having Durfee squares of sizes 1, 1, 2, 1 and 1, respectively.
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MAPLE
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g:=sum(2*i*x^(4*i^2+2*i)/product(1-x^(2*j), j=1..2*i)/product(1-x^(2*j-1), j=1..i), i=1..30)+ sum((2*i-1)*x^((2*i-1)^2)/product(1-x^(2*j), j=1..2*i-1)/product(1-x^(2*j-1), j=1..i), i=1..30): gser:=series(g, x=0, 62): seq(coeff(gser, x^n), n=1..60);
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CROSSREFS
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Cf. A116464.
Adjacent sequences: A116462 A116463 A116464 this_sequence A116466 A116467 A116468
Sequence in context: A084783 A129838 A032153 this_sequence A117356 A017819 A050044
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch and Vladeta Jovovic (deutsch(AT)duke.poly.edu), Feb 18 2006
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