A027200 Triangular array T read by rows: T(n,k) = number of partitions of n into an even number of parts, each >=k.

0, 1, 0, 1, 0, 0, 3, 1, 0, 0, 3, 1, 0, 0, 0, 6, 2, 1, 0, 0, 0, 7, 2, 1, 0, 0, 0, 0, 12, 4, 2, 1, 0, 0, 0, 0, 14, 4, 2, 1, 0, 0, 0, 0, 0, 22, 6, 3, 2, 1, 0, 0, 0, 0, 0, 27, 7, 3, 2, 1, 0, 0, 0, 0, 0, 0, 40, 11, 5, 3, 2, 1, 0, 0, 0, 0, 0, 0, 49, 12, 5, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 69, 17, 7, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Examples

			 Triangle begins:
   0;
   1,  0;
   1,  0, 0;
   3,  1, 0, 0;
   3,  1, 0, 0, 0;
   6,  2, 1, 0, 0, 0;
   7,  2, 1, 0, 0, 0, 0;
  12,  4, 2, 1, 0, 0, 0, 0;
  14,  4, 2, 1, 0, 0, 0, 0, 0;
  22,  6, 3, 2, 1, 0, 0, 0, 0, 0;
  27,  7, 3, 2, 1, 0, 0, 0, 0, 0, 0;
  40, 11, 5, 3, 2, 1, 0, 0, 0, 0, 0, 0;
  49, 12, 5, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0;
		

Crossrefs

Cf. A027186, A027187 (1st column), A027188, A027189, A027190, A027191, A027192.

Programs

PARI

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

Formula

T(n, k) = Sum{E(n, i)}, k<=i<=n, E given by A027186.

T(n,k) + A027199(n,k) = A026807(n,k). - R. J. Mathar, Oct 18 2019

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

Extensions

More terms from Seiichi Manyama, May 15 2023