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A180291
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Number of arrangements of n indistinguishable balls in n boxes with the maximum number of balls in any box equal to n-1.
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5
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1, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550
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OFFSET
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2,2
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COMMENTS
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To clarify a slight ambiguity in the definition, the heaviest box in such an arrangement should contain exactly n-1 balls. - Gus Wiseman, Sep 22 2016
Clearly a(2)=1. Moreover, for n>2, a(n) = n*(n-1), since one can choose the box with n-1 balls in n ways, and the remaining ball can be put in one of the remaining n-1 boxes. So the conjecture above and the empirical formulas below are all correct. - Luca Ferrigno, Jul 13 2023
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LINKS
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FORMULA
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Empirical: a(n) = n*binomial(n-1,n-2) for n > 2.
Empirical: a(n) = n^2 - n for n > 2. a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5. G.f.: x^2*(1 + 3*x - 3*x^2 + x^3)/(1-x)^3. - Colin Barker, Mar 18 2012
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, 0, Sum[b[n - j, i - 1, k], {j, 0, Min[n, k]}]]];
a[n_] := b[n, n, n - 1] - b[n, n, n - 2];
a[n_] := If[n == 2, 1, n*(n - 1)] (* Luca Ferrigno, Jul 13 2023 *)
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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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