A242086 - OEIS

%I #15 Feb 18 2015 03:55:01

%S 1,0,1,0,1,0,0,1,0,1,0,2,0,1,0,0,3,0,1,0,1,0,5,0,2,0,1,0,0,8,0,3,0,1,

%T 0,1,0,13,0,5,0,2,0,1,0,0,21,0,8,0,3,0,1,0,1,0,34,0,13,0,5,0,2,0,1,0,

%U 0,55,0,21,0,8,0,3,0,1,0,1,0,89,0,34,0,13,0,5,0,2,0,1,0,0,144,0,55,0,21,0,8,0,3,0,1,0,1

%N Triangle read by rows: T(n,k) is the number of compositions of n into odd parts with first part k.

%H Joerg Arndt, <a href="http://arxiv.org/abs/1405.6503">Subset-lex: did we miss an order?</a>, arXiv:1405.6503 [math.CO], (26-May-2014)

%F T(0,0)=1, T(0,k)=0 for k != 1 (first column).

%F T(n,k) = 0 for k>=1 and even.

%F T(n,n) = 1 for n>=1 and odd, otherwise T(n,n)=0.

%F T(n,k) = F(n-k-1) for n>=1 and odd k>=1, F = A000045.

%e Triangle starts:

%e 00:  1,

%e 01:  0, 1,

%e 02:  0, 1, 0,

%e 03:  0, 1, 0, 1,

%e 04:  0, 2, 0, 1, 0,

%e 05:  0, 3, 0, 1, 0, 1,

%e 06:  0, 5, 0, 2, 0, 1, 0,

%e 07:  0, 8, 0, 3, 0, 1, 0, 1,

%e 08:  0, 13, 0, 5, 0, 2, 0, 1, 0,

%e 09:  0, 21, 0, 8, 0, 3, 0, 1, 0, 1,

%e 10:  0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0,

%e 11:  0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,

%e 12:  0, 89, 0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0,

%e 13:  0, 144, 0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,

%e 14:  0, 233, 0, 89, 0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0,

%e 15:  0, 377, 0, 144, 0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,

%p b:= proc(n) b(n):=`if`(n=0, 1, add(b(n-2*j-1), j=0..(n-1)/2)) end:

%p T:= (n, k)-> `if`([n, k]=[0$2], 1, `if`(irem(k, 2)=0 or k>n, 0, b(n-k))):

%p seq(seq(T(n, k), k=0..n), n=0..15);  # _Alois P. Heinz_, May 10 2014

%t b[n_] := If[n == 0, 1, Sum[b[n-2*j-1], {j, 0, (n-1)/2}]]; T[n_, k_] := If[{n, k} == {0, 0}, 1, If[Mod[k, 2] == 0 || k>n, 0, b[n-k]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Feb 18 2015, after _Alois P. Heinz_ *)

%Y Cf. A000045.

%K nonn,tabl

%O 0,12

%A _Joerg Arndt_, May 04 2014