A027356 Array read by rows: T(n,k) = number of partitions of n into distinct odd parts in which k is the greatest part, for k=1,2,...,n, n>=1.

1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Author

Clark Kimberling, revised Jul 23 2004

Keywords

• nonn
• tabl

Comments

First T(n,k) not 0 or 1 is T(17,9)=2, which counts 1+7+9 and 3+5+9. Row sums: A000700.

Examples

			First 5 rows:
1
0 0
0 0 1
0 0 1 0
0 0 0 0 1
Row 40 with even-numbered terms deleted:
0 0 0 0 0 0 2 5 6 7 6 5 4 3 2 1 1 1 1;
E.g. final 2 counts these two partitions: 9+31 and 1+3+5+31.
		

Links

• Alois P. Heinz, Rows n = 1..361, flattened
• Sean A. Irvine, Java program (github)

Crossrefs

Cf. A000700.

T(4n+1,2n+1) gives A069910.

Programs

Maple

b:= proc(n, i) option remember; `if`(n>i^2, 0, `if`(n=0, 1,
      b(n, i-1) +(p-> `if`(p>n, 0, b(n-p, i-1)))((2*i-1))))
    end:
T:= (n, k)-> `if`(k::even, 0, b(n-k, (k-1)/2)):
seq(seq(T(n, k), k=1..n), n=1..20);  # Alois P. Heinz, Oct 28 2019

Mathematica

b[n_, i_] := b[n, i] = If[n > i^2, 0, If[n == 0, 1, b[n, i - 1] + Function[p, If[p > n, 0, b[n - p, i - 1]]][2i - 1]]];
T [n_, k_] := If[EvenQ[k], 0, b[n - k, (k - 1)/2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

Formula

T(n, 1)=0 for all n; T(n, n)=1 for all odd n>1; and for n>=3, T(n, k)=0 if k is even, else T(n, k)=Sum{T(n-k, i): i=1, 2, ..., n-1} for k=2, 3, ..., n-1.

Extensions

Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar