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A116503
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Sum of the areas of the Durfee squares of all partitions of n.
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3
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1, 2, 3, 8, 13, 26, 39, 64, 98, 148, 216, 322, 455, 648, 904, 1258, 1711, 2336, 3128, 4198, 5548, 7330, 9569, 12496, 16146, 20836, 26674, 34098, 43273, 54846, 69072, 86848, 108627, 135612, 168527, 209066, 258271, 318482, 391321, 479946, 586709
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OFFSET
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1,2
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COMMENTS
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a(n) = sum(k^2*A115994(n,k), k=1..floor(sqrt(n))).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..1000
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FORMULA
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G.f.: sum(k^2*z^(k^2)/product((1-z^j)^2, j=1..k), k=1..infinity).
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EXAMPLE
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a(4) = 8 because the partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], have Durfee squares of sizes 1,1,2,1 and 1, respectively and 1^2+1^2+2^2+1^2+1^2=8.
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MAPLE
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g:=sum(k^2*z^(k^2)/product((1-z^j)^2, j=1..k), k=1..10): gser:=series(g, z=0, 52): seq(coeff(gser, z^n), n=1..45);
# second Maple program
b:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
end:
a:= n-> add (k^2*add (b(m, k)*b(n-k^2-m, k),
m=0..n-k^2), k=1..floor(sqrt(n))):
seq (a(n), n=1..40); # Alois P. Heinz, Apr 09 2012
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CROSSREFS
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Cf. A115994, A115995.
Sequence in context: A004138 A213046 A221181 * A105204 A045692 A103196
Adjacent sequences: A116500 A116501 A116502 * A116504 A116505 A116506
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KEYWORD
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easy,nonn,changed
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AUTHOR
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Emeric Deutsch, Vladeta Jovovic, Feb 18 2006
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STATUS
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approved
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