A027189 -id:A027189 - cp's OEIS frontend

A027192 Number of partitions of n into an odd number of parts, the least being 6; also, a(n+6) = number of partitions of n into an even number of parts, each >=6.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 14, 15, 19, 21, 26, 29, 35, 39, 48, 53, 63, 71, 84, 94, 111, 124, 145, 163, 189, 212, 247, 276, 318, 358, 411, 461, 529, 593, 678, 761, 866, 971, 1106, 1238, 1404, 1574, 1781, 1993

Offset: 1

Clark Kimberling

nonn

In general, if m>=1 and g.f. = x^m * Sum_{k>=0} x^(2*m*k) / Product_{j=1..2*k} (1-x^j), then a(n) ~ Pi^(m-1) * (m-1)! * exp(Pi*sqrt(2*n/3)) / (2^((m+5)/2) * 3^(m/2) * n^((m+1)/2)). - Vaclav Kotesovec, Jun 20 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..5000

Cf. A027187, A027188, A027189, A027190, A027191, A027198.

nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 - x^(2*k))*(1 - x^(2*k - 1))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += x^(12*k)/p;, {k, 1, nmax}]; Join[{0, 0, 0, 0, 0}, CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 20 2025 *)

G.f.: x^6 * Sum_{k>=0} x^(12*k)/Product_{j=1..2*k} (1-x^j). - Seiichi Manyama, May 15 2023

a(n) ~ 5 * Pi^5 * exp(Pi*sqrt(2*n/3)) / (9 * 2^(5/2) * n^(7/2)). - Vaclav Kotesovec, Jun 20 2025

More terms from Vladeta Jovovic, Aug 01 2009

A027195 Number of partitions of n into an even number of parts, the least being 3; also, a(n+3) = number of partitions of n into an odd number of parts, each >=3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 16, 20, 24, 31, 36, 45, 54, 66, 78, 97, 114, 138, 164, 197, 232, 280, 328, 392, 461, 546, 639, 758, 884, 1041, 1215, 1425, 1657, 1941, 2250, 2624, 3041, 3534, 4084, 4740, 5465, 6321, 7280, 8399

Offset: 1

Clark Kimberling

nonn

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

Cf. A026796, A027189.

b:= proc(n, i, t) option remember; `if`(n=0, t,
     `if`(i>n, 0, b(n, i+1, t)+b(n-i, i, 1-t)))
    end:
a:= n-> `if`(n<3, 0, b(n-3, 3, 0)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 18 2019

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i, 1 - t]]];
a[n_] := If[n < 3, 0, b[n - 3, 3, 0]];
Array[a, 100] (* Jean-François Alcover, May 17 2020, after Alois P. Heinz *)

a(n) + A027189(n) = A026796(n). - R. J. Mathar, Oct 18 2019
a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (8 * 3^(3/2) * n^2). - Vaclav Kotesovec, May 17 2020
G.f.: x^6 * Sum_{k>=0} x^(6*k)/Product_{j=1..2*k+1} (1-x^j). - Seiichi Manyama, May 15 2023

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

A027192 Number of partitions of n into an odd number of parts, the least being 6; also, a(n+6) = number of partitions of n into an even number of parts, each >=6.

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A027195 Number of partitions of n into an even number of parts, the least being 3; also, a(n+3) = number of partitions of n into an odd number of parts, each >=3.

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

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