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A180291 Number of arrangements of n indistinguishable balls in n boxes with the maximum number of balls in any box equal to n-1.

%I A180291 #56 Aug 28 2025 06:35:36
%S A180291 1,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,
%T A180291 462,506,552,600,650,702,756,812,870,930,992,1056,1122,1190,1260,1332,
%U A180291 1406,1482,1560,1640,1722,1806,1892,1980,2070,2162,2256,2352,2450,2550
%N A180291 Number of arrangements of n indistinguishable balls in n boxes with the maximum number of balls in any box equal to n-1.
%C A180291 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
%C A180291 Conjecture: For n > 2, a(n) = 2*A000217(n+1). - _Bruce J. Nicholson_, Apr 05 2017
%C A180291 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
%C A180291 a(n) is the maximum number of squares that can be marked on an n X n bingo board without getting bingo (i.e., without completing any row, column, or diagonal). - _Rasmus Lindahl_, Feb 23 2025
%H A180291 R. H. Hardin, <a href="/A180291/b180291.txt">Table of n, a(n) for n = 2..59</a>
%F A180291 Empirical: a(n) = n*binomial(n-1,n-2) for n > 2.
%F A180291 Empirical: a(n) = A002378(n-1) for n > 2. - _R. J. Mathar_, Sep 06 2010
%F A180291 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
%F A180291 a(n) = n*(n-1) for n > 2. - _Luca Ferrigno_, Jul 13 2023
%t A180291 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]}]]];
%t A180291 a[n_] := b[n, n, n - 1] - b[n, n, n - 2];
%t A180291 Table[a[n], {n, 2, 51}] (* _Jean-François Alcover_, Aug 28 2022, after _Alois P. Heinz_ in A180281 *)
%t A180291 a[n_] := If[n == 2, 1, n*(n - 1)] (* _Luca Ferrigno_, Jul 13 2023 *)
%Y A180291 (n-1)-th entry in rows of A180281.
%Y A180291 Essentially the same as A060798 and A002378.
%K A180291 nonn,changed
%O A180291 2,2
%A A180291 _R. H. Hardin_, formula from _Robert Gerbicz_ in the Sequence Fans Mailing List, Aug 24 2010