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

1,3

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).

Links

Table of n, a(n) for n=1..55.

Formula

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

Example

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
  ...

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 *)

Crossrefs

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

Sequence in context: A217234 A051290 A227338 * A144878 A049424 A157394

Adjacent sequences: A125102 A125103 A125104 * A125106 A125107 A125108

Keyword

easy,nonn,tabl

Author

Alford Arnold, Dec 07 2006

Extensions

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

Status

approved