A382820 -id:A382820 - cp's OEIS frontend

A382818 Square array A(n,k), n > 0, k > 0, read by downward antidiagonals: A(n,k) is the number of columns in all k-compositions of n.

1, 2, 3, 3, 11, 8, 4, 24, 52, 20, 5, 42, 163, 227, 48, 6, 65, 372, 1017, 944, 112, 7, 93, 710, 3019, 6030, 3800, 256, 8, 126, 1208, 7095, 23256, 34563, 14944, 576, 9, 164, 1897, 14340, 67251, 173076, 193392, 57748, 1280, 10, 207, 2808, 26082, 161394, 615630, 1256936, 1062756, 220128, 2816

Offset: 1

John Tyler Rascoe, Apr 05 2025

A k-composition of n is a rectangular array of nonnegative integers with k rows, at least one nonzero entry in each column, and having the sum of all entries equal to n.

			Square array begins:
   1,   2,    3,     4,     5,      6, ...
   3,  11,   24,    42,    65,     93, ...
   8,  52,  163,   372,   710,   1208, ...
  20, 227, 1017,  3019,  7095,  14340, ...
  48, 944, 6030, 23256, 67251, 161394, ...
  ...
A(2,2) = 11 counts the columns in the 2-compositions of 2:
 [2]   [0]   [1]   [1,0]   [0,1]   [0,0]   [1,1]
 [0],  [2],  [1],  [0,1],  [1,0],  [1,1],  [0,0].

John Tyler Rascoe, Antidiagonals n = 1..100, flattened

C.f. A001792 (column k=1), A005475 (row n=2), A145839, A181289, A181290 (column k=2), A382820 (main diagonal).

A382818_Column(k,N) = {my(x='x+O('x^N)); Vec(-(((1 - x)^k - 1)*(1 - x)^k)/( ((1 - x)^k - 1) + (1 - x)^k)^2)}
A382818_array(max_row) = {my(m=matrix(0)); for(n=1,max_row, m=matconcat([m,A382818_Column(n,max_row)~])); m}
A382818_array(10)

Column k has g.f.: -((1 - x)^k - 1)*(1 - x)^k/(((1 - x)^k - 1) + (1 - x)^k)^2.

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A382818 Square array A(n,k), n > 0, k > 0, read by downward antidiagonals: A(n,k) is the number of columns in all k-compositions of n.

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