A242086 Triangle read by rows: T(n,k) is the number of compositions of n into odd parts with first part k.

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, 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, 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

Offset

0,12

Links

Table of n, a(n) for n=0..104.

Joerg Arndt, Subset-lex: did we miss an order?, arXiv:1405.6503 [math.CO], (26-May-2014)

Formula

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

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

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

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

Example

Triangle starts:
00:  1,
01:  0, 1,
02:  0, 1, 0,
03:  0, 1, 0, 1,
04:  0, 2, 0, 1, 0,
05:  0, 3, 0, 1, 0, 1,
06:  0, 5, 0, 2, 0, 1, 0,
07:  0, 8, 0, 3, 0, 1, 0, 1,
08:  0, 13, 0, 5, 0, 2, 0, 1, 0,
09:  0, 21, 0, 8, 0, 3, 0, 1, 0, 1,
10:  0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0,
11:  0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,
12:  0, 89, 0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0,
13:  0, 144, 0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,
14:  0, 233, 0, 89, 0, 34, 0, 13, 0, 5, 0, 2, 0, 1, 0,
15:  0, 377, 0, 144, 0, 55, 0, 21, 0, 8, 0, 3, 0, 1, 0, 1,

Maple

b:= proc(n) b(n):=`if`(n=0, 1, add(b(n-2*j-1), j=0..(n-1)/2)) end:
T:= (n, k)-> `if`([n, k]=[0$2], 1, `if`(irem(k, 2)=0 or k>n, 0, b(n-k))):
seq(seq(T(n, k), k=0..n), n=0..15);  # Alois P. Heinz, May 10 2014

Mathematica

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 *)

Crossrefs

Cf. A000045.

Sequence in context: A355619 A355607 A253184 * A160973 A036853 A036852

Adjacent sequences: A242083 A242084 A242085 * A242087 A242088 A242089

Keyword

nonn,tabl

Author

Joerg Arndt, May 04 2014

Status

approved