A125105 Triangular array with the first half of the odd-indexed rows of A048004.

1, 1, 4, 1, 12, 11, 1, 33, 47, 27, 1, 88, 185, 127, 63, 1, 232, 694, 563, 303, 143, 1, 609, 2526, 2400, 1394, 687, 319, 1, 1596, 9012, 9960, 6215, 3186, 1519, 703, 1, 4180, 31709, 40534, 27095, 14401, 7026, 3311, 1535, 1, 10945, 110469, 162538, 116143, 63872, 31808, 15218, 7151, 3327

Offset: 1

Author

Alford Arnold, Dec 07 2006

Keywords

• easy
• nonn
• tabl

Comments

A000079 counts compositions admitting a variety of triangular views; for example, A048004 and A105147. The subtable formed from the odd rows of A048004 has row sums 1, 8, 44, 208, 912, ... . Because only the first half of rows of A048004 is transferred to this triangle here, there is a difference between row sums of A048004 and row sums here, A045623(n-1).

Examples

			The odd-indexed rows of triangle A048004 begin
  1  1
  1  4  2 1
  1 12 11 5 2 1
  ...
so the triangle here begins
  1
  1  4
  1 12 11
  ...
		

Crossrefs

Cf. A008464 (row sums), A008464, A045623, A048004, A105147, A125104.

Programs

Maple

A048004 := proc(n,k) option remember ; if k < 0 or k > n then 0; elif k = 0 or k = n then 1; else 2*procname(n-1,k)+procname(n-1,k-1)-2*procname(n-2,k-1)+procname(n-k-1,k-1)-procname(n-k-2,k) ; fi ; end:
A125105 := proc(n,k) A048004(2*n-1,k) ; end:
for n from 1 to 13 do for k from 0 to n-1 do printf("%d ",A125105(n,k)) ; od: od: # R. J. Mathar, Nov 23 2007

Mathematica

B[n_, k_] := B[n, k] = If[n == 0 || k == 1, 1, Sum[B[n - j, k], {j, 1, Min[n, k]}]];
A048004[n_, k_] := B[n + 1, k + 1] - B[n + 1, k];
T[n_, k_] := A048004[2 n - 1, k];
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 27 2024, after Maple code here and in A048004 *)

Formula

T(n,k) = A048004(2*n-1,k), 0 <= k < n. - R. J. Mathar, Nov 23 2007

Extensions

More terms from R. J. Mathar, Nov 23 2007