A027186 -id:A027186 - cp's OEIS frontend

A027185 Triangular array O by rows: O(n,k) = number of partitions of n into an odd number of parts, the least being k.

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 6, 1, 0, 0, 0, 0, 1, 7, 2, 0, 0, 0, 0, 0, 1, 12, 2, 1, 0, 0, 0, 0, 0, 1, 14, 4, 1, 0, 0, 0, 0, 0, 0, 1, 22, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1, 27, 6, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 40, 7, 3, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling

			1,
0, 1,
1, 0, 1,
1, 0, 0, 1,
3, 0, 0, 0, 1,
3, 1, 0, 0, 0, 1,
6, 1, 0, 0, 0, 0, 1,
7, 2, 0, 0, 0, 0, 0, 1,
12, 2, 1, 0, 0, 0, 0, 0, 1,
14, 4, 1, 0, 0, 0, 0, 0, 0, 1,
22, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1,
27, 6, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1,
40, 7, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,

Cf. A027186.

O(n, k) = E(n-k, k)+E(n-k, k+1)+...+E(n-k, n-k), for 2<=2k<=n, E given by A027186.

T(n,k) + A027186(n,k) = A026794(n,k). - R. J. Mathar, Oct 18 2019

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

Original entry on oeis.org

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

Clark Kimberling

			 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;

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

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

 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

More terms from Seiichi Manyama, May 15 2023

cp's OEIS Frontend

A027185 Triangular array O by rows: O(n,k) = number of partitions of n into an odd number of parts, the least being k.

Views

Author

Keywords

Examples

Crossrefs

Formula