A027185 -id:A027185 - cp's OEIS frontend

A027186 Triangular array E by rows: E(n,k) = number of partitions of n into an even number of parts, the least being k.

0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 4, 1, 1, 0, 0, 0, 5, 1, 1, 0, 0, 0, 0, 8, 2, 1, 1, 0, 0, 0, 0, 10, 2, 1, 1, 0, 0, 0, 0, 0, 16, 3, 1, 1, 1, 0, 0, 0, 0, 0, 20, 4, 1, 1, 1, 0, 0, 0, 0, 0, 0, 29, 6, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 37, 7, 2, 1, 1, 1, 0, 0, 0, 0

Offset: 1

Clark Kimberling

			 0,
 1, 0,
 1, 0, 0,
 2, 1, 0, 0,
 2, 1, 0, 0, 0,
 4, 1, 1, 0, 0, 0,
 5, 1, 1, 0, 0, 0, 0,
 8, 2, 1, 1, 0, 0, 0, 0,
10, 2, 1, 1, 0, 0, 0, 0, 0,
16, 3, 1, 1, 1, 0, 0, 0, 0, 0,
20, 4, 1, 1, 1, 0, 0, 0, 0, 0, 0,
29, 6, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0,
37, 7, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0,

Cf. A027185.

E(n, k) = O(n-k, k)+O(n-k, k+1)+...+O(n-k, n-k), for 2<=2k<=n, O given by A027185.

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

Original entry on oeis.org

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

Clark Kimberling

			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;

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

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

 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

More terms from Seiichi Manyama, May 15 2023

cp's OEIS Frontend

A027186 Triangular array E by rows: E(n,k) = number of partitions of n into an even number of parts, the least being k.

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