A027195 -id:A027195 - cp's OEIS frontend

A027189 Number of partitions of n into an odd number of parts, the least being 3; also, a(n+3) = number of partitions of n into an even number of parts, each >=3.

0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 17, 19, 25, 29, 37, 43, 56, 64, 80, 94, 116, 135, 167, 194, 236, 276, 332, 387, 466, 541, 645, 752, 891, 1034, 1223, 1417, 1666, 1931, 2260, 2613, 3054, 3522, 4098, 4725, 5480

Offset: 1

Clark Kimberling

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A027195.

a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (8 * 3^(3/2) * n^2). - Vaclav Kotesovec, May 17 2020

G.f.: x^3 * Sum_{k>=0} x^(6*k)/Product_{j=1..2*k} (1-x^j). - Seiichi Manyama, May 15 2023

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

A027189 Number of partitions of n into an odd number of parts, the least being 3; also, a(n+3) = number of partitions of n into an even number of parts, each >=3.

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

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