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A354468
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Number of possible ordered pairs (n_1, S) where (n_1, n_2, ..., n_k) is a partition of n, n_1 is the largest element of the partition, and S = Sum_{j=1..k} n_j^2.
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3
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1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 39, 50, 66, 83, 104, 127, 157, 188, 225, 265, 312, 359, 418, 479, 547, 620, 700, 786, 884, 987, 1094, 1214, 1348, 1479, 1627, 1779, 1945, 2122, 2313, 2505, 2719, 2934, 3161, 3412, 3666, 3932, 4218, 4511, 4820, 5140, 5477, 5825
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OFFSET
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0,3
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COMMENTS
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In categorical data with a sample of size n distributed over at least 1 and at most n distinct categorical types, if a dataset is summarized by an ordered pair of two numbers -- the number of observations of the most frequent type and the sum of squares of the frequencies of all types -- then a(n) gives the number of distinguishable ordered pairs across all possible datasets.
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LINKS
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EXAMPLE
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For n=4 the a(4)=5 ordered pairs are (4,16), (3,10), (2,8), (2,6), and (1,4).
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=1, {n},
{b(n, i-1)[], map(x-> x+i^2, b(n-i, min(n-i, i)))[]})
end:
a:= n-> add(nops(b(n-i, min(n-i, i))), i=signum(n)..n):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {n},
Union@Flatten@{b[n, i-1], #+i^2& /@ b[n-i, Min[n-i, i]]}];
a[n_] := Sum[Length[b[n-i, Min[n-i, i]]], {i, Sign[n], n}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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