A027199 Triangular array T read by rows: T(n,k) = number of partitions of n into an odd number of parts, each >=k.

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 8, 2, 1, 1, 1, 1, 1, 10, 3, 1, 1, 1, 1, 1, 1, 16, 4, 2, 1, 1, 1, 1, 1, 1, 20, 6, 2, 1, 1, 1, 1, 1, 1, 1, 29, 7, 3, 1, 1, 1, 1, 1, 1, 1, 1, 37, 10, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 52, 12, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 66, 17, 6, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Examples

			Triangle begins:
   1;
   1,  1;
   2,  1, 1;
   2,  1, 1, 1;
   4,  1, 1, 1, 1;
   5,  2, 1, 1, 1, 1;
   8,  2, 1, 1, 1, 1, 1;
  10,  3, 1, 1, 1, 1, 1, 1;
  16,  4, 2, 1, 1, 1, 1, 1, 1;
  20,  6, 2, 1, 1, 1, 1, 1, 1, 1;
  29,  7, 3, 1, 1, 1, 1, 1, 1, 1, 1;
  37, 10, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1;
  52, 12, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
		

Crossrefs

Cf. A027185, A027193 (first column), A027194, A027195, A027196, A027197, A027198.

Programs

PARI

T(n, k) = polcoef(x^k*sum(i=0, n, x^(2*k*i)/prod(j=1, 2*i+1, 1-x^j+x*O(x^n))), n); \\ Seiichi Manyama, May 15 2023

Formula

T(n, k) = Sum{O(n, i)}, k<=i<=n, O given by A027185.

T(n,k) + A027200(n,k) = A026807(n,k). - R. J. Mathar, Oct 18 2019

G.f. of column k: x^k * Sum_{i>=0} x^(2*k*i)/Product_{j=1..2*i+1} (1-x^j). - Seiichi Manyama, May 15 2023

Extensions

More terms from Seiichi Manyama, May 15 2023