A382923 -id:A382923 - cp's OEIS frontend

A382924 Number of m-compositions of n with n zeros.

1, 2, 13, 70, 336, 2076, 11091, 65210, 365661, 2159354, 11713047, 71427504, 392916687, 2245186352, 13527678851, 73679458270, 429472428457, 2553994191220, 14264421153074, 80483620074092, 489077890675807, 2768919905996888, 15394229582049408, 91794448088043258

Offset: 0

John Tyler Rascoe, Apr 09 2025

nonn

For some m > 0, an m-composition of n is a rectangular array of nonnegative integers with m rows, at least one nonzero entry in each column, and having the sum of all entries equal to n.

			a(2) = 13 counts:
  [2]  [0]  [0]  [1]  [1]  [1]  [0]  [0]  [0]  [1][1]  [1][0]  [0][0]  [0][1]
  [0]  [2]  [0]  [1]  [0]  [0]  [1]  [1]  [0]  [0][0], [0][1], [1][1], [1][0].
  [0], [0], [2], [0]  [1]  [0]  [1]  [0]  [1]
                 [0], [0], [1], [0], [1], [1],

Cf. A038207, A101509, A181331, A261780, A323429, A382820, (main diagonal of A382923).

G_tx(max_row) = {my(row = max_row, N = row*2, m = List([concat([1],vector(row-1,i,0))]), x='x+O('x^N), h=1 + sum(m=1,N,-1+ 1/(1 + t^m - (t + x/(1-x))^m))); for(n=1,row, listput(m,Vecrev(polcoeff(h, n))[1..row])); matrix(row, row, i,j, m[i][j])}
A382924(max_n) ={my(A=G_tx(max_n)); vector(max_n,i,A[i,i])}
A382924(20)

a(n) = [(x*t)^n] 1 + Sum_{m>0} -1 + 1/(1 + t^m - (t + x/(1 - x))^m).

A383256 Number of n X n matrices of nonnegative entries with all columns summing to n and no horizontally adjacent zeros.

Original entry on oeis.org

1, 1, 7, 343, 125465, 366908001, 8698468668251, 1708834003295306868, 2810884261025802145414705, 39088555382409783097546399456477, 4626844513673581956954679383115038810744, 4688191496359773864437279635019555242588548880831

Offset: 0

John Tyler Rascoe, Apr 21 2025

nonn

			a(1) = 1: [1]
a(2) = 7: [1,1]   [1,0]   [1,2]   [0,1]   [2,1]   [0,2]   [2,0]
          [1,1],  [1,2],  [1,0],  [2,1],  [0,1],  [2,0],  [0,2].

John Tyler Rascoe, Python program.

Cf. A008300, A120733, A145839, A261780, A382923.

Python
```
# see links
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a(10)-a(11) from Bert Dobbelaere, Apr 23 2025

cp's OEIS Frontend

A382924 Number of m-compositions of n with n zeros.

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