A123685 Counts compositions as described by table A047969; however, only those ending with an odd part are considered.

1, 1, 0, 1, 1, 1, 1, 3, 4, 0, 1, 7, 14, 2, 1, 1, 15, 46, 14, 7, 0, 1, 31, 146, 74, 43, 3, 1, 1, 63, 454, 350, 247, 33, 10, 0, 1, 127, 1394, 1562, 1363, 273, 88, 4, 1, 1, 255, 4246, 6734, 7327, 2013, 724, 60, 13, 0, 1, 511, 12866, 28394, 38683, 13953, 5716, 676, 149, 5, 1, 1

Offset: 1

Author

Alford Arnold, Oct 11 2006

Keywords

• nonn
• tabl

Examples

			Row four of table A047969 counts the 14 compositions
4
31 13 32 23 33
211 121 112 221 212 122 222
1111
whereas A123685 only counts
31 13 32 33
121 112 122
and 1111
		

Links

• Alois P. Heinz, Antidiagonals n = 1..141, flattened

Crossrefs

Diagonals include A000012, A059841, A000225, A123684 and A027649.

Programs

Maple

g:= proc(b, t, l, m) option remember; `if`(t=0, b*l, add(
      g(b, t-1, irem(k, 2), m), k=1..m-1)+g(1, t-1, irem(m, 2), m))
    end:
A:= (n, k)-> g(0, k, 0, n):
seq(seq(A(n, d+1-n), n=1..d), d=1..13); # Alois P. Heinz, Nov 06 2009

Mathematica

g[b_, t_, l_, m_] := g[b, t, l, m] = If[t == 0, b*l, Sum[g[b, t-1, Mod[k, 2], m], {k, 1, m-1}] + g[1, t-1, Mod[m, 2], m]]; A[n_, k_] := g[0, k, 0, n]; Table [Table [A[n, d+1-n], {n, 1, d}], {d, 1, 13}] // Flatten (* Jean-François Alcover, Feb 20 2015, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Nov 06 2009