A382923 Square array A(n,k), n >= 0, k >= 0, read by downward antidiagonals: A(n,k) is the number of m-compositions of n with k zeros.

1, 0, 1, 0, 2, 3, 0, 3, 5, 7, 0, 4, 13, 16, 16, 0, 5, 14, 33, 40, 35, 0, 6, 29, 70, 105, 100, 75, 0, 7, 27, 88, 207, 292, 244, 159, 0, 8, 51, 152, 336, 604, 758, 576, 334, 0, 9, 44, 206, 588, 1161, 1749, 1920, 1329, 696, 0, 10, 79, 300, 882, 2076, 3685, 4924, 4802, 3028, 1442

Offset: 0

Author

John Tyler Rascoe, Apr 09 2025

Keywords

• nonn
• easy
• tabl

Comments

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.

Examples

			Square array begins:
   1,   0,   0,   0,    0,    0, ...
   1,   2,   3,   4,    5,    6, ...
   3,   5,  13,  14,   29,   27, ...
   7,  16,  33,  70,   88,  152, ...
  16,  40, 105, 207,  336,  588, ...
  35, 100, 292, 604, 1161, 2076, ...
  ...
A(2,0) = 3 counts:
  [2],  [1,1],  [1]
                [1].
A(2,1) = 5 counts:
  [2]   [0]   [1]   [1]   [0]
  [0],  [2],  [1]   [0]   [1]
              [0],  [1],  [1].
		

Links

• John Tyler Rascoe, Antidiagonals n = 0..30, flattened

Crossrefs

Cf. A038207, A101509 (column k=0), A181331, A261780, A323429, A382924 (main diagonal).

Programs

PARI

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])}
G_tx(10)

Formula

G.f.: G(t,x) = 1 + Sum_{m>0} -1 + 1/(1 + t^m - (t + x/(1 - x))^m).