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

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

Offset: 2

Author

R. H. Hardin, formula from Robert Gerbicz in the Sequence Fans Mailing List, Aug 24 2010

Keywords

• nonn
• changed

Comments

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

Conjecture: For n > 2, a(n) = 2*A000217(n+1). - Bruce J. Nicholson, Apr 05 2017

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

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

Links

• R. H. Hardin, Table of n, a(n) for n = 2..59

Crossrefs

(n-1)-th entry in rows of A180281.

Essentially the same as A060798 and A002378.

Programs

Mathematica

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];
Table[a[n], {n, 2, 51}] (* Jean-François Alcover, Aug 28 2022, after Alois P. Heinz in A180281 *)
a[n_] := If[n == 2, 1, n*(n - 1)] (* Luca Ferrigno, Jul 13 2023 *)

Formula

Empirical: a(n) = n*binomial(n-1,n-2) for n > 2.

Empirical: a(n) = A002378(n-1) for n > 2. - R. J. Mathar, Sep 06 2010

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

a(n) = n*(n-1) for n > 2. - Luca Ferrigno, Jul 13 2023