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

%I A123685 #15 Feb 23 2015 20:29:43
%S A123685 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,
%T A123685 63,454,350,247,33,10,0,1,127,1394,1562,1363,273,88,4,1,1,255,4246,
%U A123685 6734,7327,2013,724,60,13,0,1,511,12866,28394,38683,13953,5716,676,149,5,1,1
%N A123685 Counts compositions as described by table A047969; however, only those ending with an odd part are considered.
%H A123685 Alois P. Heinz, <a href="/A123685/b123685.txt">Antidiagonals n = 1..141, flattened</a>
%e A123685 Row four of table A047969 counts the 14 compositions
%e A123685 4
%e A123685 31 13 32 23 33
%e A123685 211 121 112 221 212 122 222
%e A123685 1111
%e A123685 whereas A123685 only counts
%e A123685 31 13 32 33
%e A123685 121 112 122
%e A123685 and 1111
%p A123685 g:= proc(b, t, l, m) option remember; `if`(t=0, b*l, add(
%p A123685       g(b, t-1, irem(k, 2), m), k=1..m-1)+g(1, t-1, irem(m, 2), m))
%p A123685     end:
%p A123685 A:= (n, k)-> g(0, k, 0, n):
%p A123685 seq(seq(A(n, d+1-n), n=1..d), d=1..13); # _Alois P. Heinz_, Nov 06 2009
%t A123685 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_ *)
%Y A123685 Diagonals include A000012, A059841, A000225, A123684 and A027649.
%K A123685 nonn,tabl
%O A123685 1,8
%A A123685 _Alford Arnold_, Oct 11 2006
%E A123685 More terms from _Alois P. Heinz_, Nov 06 2009