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A268188
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Sum of the sizes of the Durfee squares of all no-leg partitions of n (or of all no-arm partitions of n).
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10
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1, 1, 1, 3, 3, 5, 5, 7, 10, 12, 15, 20, 23, 28, 34, 43, 49, 61, 71, 87, 100, 120, 137, 164, 190, 221, 254, 298, 340, 396, 451, 520, 592, 679, 769, 883, 996, 1133, 1278, 1453, 1632, 1850, 2072, 2339, 2620, 2947, 3288, 3695, 4119, 4608, 5129, 5728, 6360, 7091, 7862
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OFFSET
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1,4
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COMMENTS
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Given a partition P, the partition formed by the cells situated below the Durfee square of P is called the leg of P. Similarly, the partition formed by the cells situated to the right of the Durfee square of P is called the arm of P.
Conjecture: also a(n) is the number of parts in the partitions of n whose parts differ by at least 2. For example, for n = 5, these partitions are 5, 41 with 3 parts in all. George Beck, Apr 22 2017
Note added Apr 22 2017. George E. Andrews informed me that this is part of the common interpretation of the Rogers-Ramanujan identities. - George Beck
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REFERENCES
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George E. Andrews, "Partitions and Durfee Dissection", Amer. J. Math. 101(1979), 735-742.
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LINKS
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FORMULA
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G.f.: g = Sum_{k>=1} (k*x^(k^2)/Product_{i=1..k}(1-x^i)).
a(n) ~ 3^(1/4) * log(phi) * phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (2*Pi*n^(1/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 09 2020
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EXAMPLE
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a(9)=10 because the no-leg partitions of 9 are [9], [7,2], [6,3], [5,4], and [3,3,3] with sizes of Durfee squares 1,2,2,2, and 3, respectively.
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MAPLE
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g := add(k*x^(k^2)/mul(1-x^i, i = 1 .. k), k = 1 .. 100): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 1 .. 55);
# 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*b(n-k^2, k), k=1..floor(sqrt(n))):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]; a[n_] := Sum[k*b[n - k^2, k], {k, 1, Floor[Sqrt[n]]}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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