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

1,2

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.

Links

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

Formula

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

Example

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].

Prog

(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)

Crossrefs

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

Sequence in context: A379351 A210194 A210755 * A219224 A265532 A112858

Adjacent sequences: A382815 A382816 A382817 * A382819 A382820 A382821

Keyword

nonn,easy,tabl

Author

John Tyler Rascoe, Apr 05 2025

Status

approved