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

1,145

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.

Links

Alois P. Heinz, Rows n = 1..361, flattened

Sean A. Irvine, Java program (github)

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.

Example

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.

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 *)

Crossrefs

Cf. A000700.

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

Sequence in context: A173856 A288926 A373840 * A361114 A181663 A359370

Adjacent sequences: A027353 A027354 A027355 * A027357 A027358 A027359

Keyword

nonn,tabl

Author

Clark Kimberling, revised Jul 23 2004

Extensions

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

Status

approved