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

Author

John Tyler Rascoe, Apr 05 2025

Keywords

• nonn
• easy
• tabl

Comments

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.

Examples

			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].
		

Links

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

Crossrefs

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

Programs

PARI

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)

Formula

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