A318161 -id:A318161 - cp's OEIS frontend

A180281 Triangle read by rows: T(n,k) = number of arrangements of n indistinguishable balls in n boxes with the maximum number of balls in any box equal to k.

1, 1, 2, 1, 6, 3, 1, 18, 12, 4, 1, 50, 50, 20, 5, 1, 140, 195, 90, 30, 6, 1, 392, 735, 392, 147, 42, 7, 1, 1106, 2716, 1652, 672, 224, 56, 8, 1, 3138, 9912, 6804, 2970, 1080, 324, 72, 9, 1, 8952, 35850, 27600, 12825, 4950, 1650, 450, 90, 10, 1, 25652, 128865, 110715, 54450, 22022, 7865, 2420, 605, 110, 11

Offset: 1

R. H. Hardin, Aug 24 2010

To clarify a slight ambiguity in the definition, the heaviest box in such an arrangement should contain exactly k balls. - Gus Wiseman, Sep 22 2016

			The T(4,2)=18 arrangements are {0022, 0112, 0121, 0202, 0211, 0220, 1012, 1021, 1102, 1120, 1201, 1210, 2002, 2011, 2020, 2101, 2110, 2200}.
Triangle starts
  1
  1   2
  1   6   3
  1  18  12  4
  1  50  50 20  5
  1 140 195 90 30 6
  ...

Alois P. Heinz, Rows n = 1..200, flattened (first 59 rows from R. H. Hardin)

Cf. A097861, A180282, A180291.
Row sums give A088218.
T(n,ceiling(n/2)) gives A318160.
T(2n,n) gives A318161.
T(2n-1,n) gives A318161.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, 0, add(b(n-j, i-1, k), j=0..min(n, k))))
    end:
T:= (n, k)-> b(n$2, k)-b(n$2, k-1):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Aug 16 2018
# second Maple program:
T:= (n, k)-> coeff(series(((x^(k+1)-1)/(x-1))^n
             -((x^k-1)/(x-1))^n, x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Aug 17 2018

T[n_,k_]:=Select[Tuples[Range[0,k],n],And[Max[#]===k,Total[#]===n]&]; (* Gus Wiseman, Sep 22 2016 *)
SequenceForm@@@T[4,2] (* example *)
Join@@Table[Length[T[n,k]],{n,1,6},{k,1,n}] (* sequence *)
(* Second program: *)
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[n_, k_] := b[n, n, k] - b[n, n, k-1];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Aug 28 2022, after Alois P. Heinz *)

Empirical: right half of table, T(n,k) = n*binomial(2*n-k-2,n-2) for 2*k > n; also, T(n,2) = Sum_{j=1..n} binomial(n,j)*binomial(n-j,j) = 2*A097861(n). - Robert Gerbicz in the Sequence Fans Mailing List
From Alois P. Heinz, Aug 17 2018: (Start)
T(n,k) = [x^n] ((x^(k+1)-1)/(x-1))^n - ((x^k-1)/(x-1))^n.
T(n,k) = A305161(n,k) - A305161(n,k-1). (End)

A318160 Number of compositions of n into exactly n nonnegative parts with largest part ceiling(n/2).

Original entry on oeis.org

1, 1, 1, 6, 18, 50, 195, 392, 1652, 2970, 12825, 22022, 96030, 160888, 705341, 1162800, 5116200, 8335338, 36773397, 59366450, 262462010, 420630210, 1862790699, 2967563040, 13160496684, 20861295000, 92624149475, 146203657992, 649794035142, 1021964428880

Offset: 0

Alois P. Heinz, Aug 19 2018

nonn

			a(3) = 6: 012, 021, 102, 120, 201, 210.
a(4) = 18: 0022, 0112, 0121, 0202, 0211, 0220, 1012, 1021, 1102, 1120, 1201, 1210, 2002, 2011, 2020, 2101, 2110, 2200.
a(5) = 50: 00023, 00032, 00113, 00131, 00203, 00230, 00302, 00311, 00320, 01013, 01031, 01103, 01130, 01301, 01310, 02003, 02030, 02300, 03002, 03011, 03020, 03101, 03110, 03200, 10013, 10031, 10103, 10130, 10301, 10310, 11003, 11030, 11300, 13001, 13010, 13100, 20003, 20030, 20300, 23000, 30002, 30011, 30020, 30101, 30110, 30200, 31001, 31010, 31100, 32000.

Alois P. Heinz, Table of n, a(n) for n = 0..2409

Bisections give: A318161 (even part), A318162 (odd part).

Cf. A180281.

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_] := If[n == 0, 1, b[n, n, Ceiling[n/2]] - b[n, n, Ceiling[n/2] - 1]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 28 2022, after Alois P. Heinz in A180281 *)

a(n) = A180281(n,ceiling(n/2)).

a(n) = 3^(3*n/2 - 7/4 + (-1)^n/4) * sqrt(n/Pi) / 2^(n - 3/2). - Vaclav Kotesovec, Sep 21 2019

cp's OEIS Frontend

A180281 Triangle read by rows: T(n,k) = number of arrangements of n indistinguishable balls in n boxes with the maximum number of balls in any box equal to k.

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