A027188 -id:A027188 - 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

A027194 Number of partitions of n into an even number of parts, the least being 2; also, a(n+2) = number of partitions of n into an odd number of parts, each >=2.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 17, 21, 27, 33, 44, 53, 68, 83, 105, 127, 159, 192, 239, 288, 353, 424, 519, 620, 752, 898, 1083, 1288, 1545, 1831, 2188, 2587, 3074, 3624, 4294, 5046, 5954, 6982, 8210, 9601, 11255, 13129

Offset: 1

Clark Kimberling

nonn

a(n+2) + A027188(n+2) = A002865(n). - R. J. Mathar, Jun 16 2018

G.f.: x^4 * Sum_{k>=0} x^(4*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

A298596 Expansion of Product_{k>=2} 1/(1 + x^k).

Original entry on oeis.org

1, 0, -1, -1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0, -1, 1, -1, 0, -1, 2, -1, 0, -2, 2, -1, 1, -2, 3, -2, 1, -3, 4, -2, 2, -4, 5, -3, 3, -5, 6, -5, 4, -6, 9, -6, 5, -9, 10, -8, 8, -11, 13, -11, 10, -14, 17, -14, 13, -19, 21, -18, 18, -23, 26, -24, 23, -29, 34, -30, 29, -38, 41, -38, 39

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

sign

The difference between the number of partitions of n into an even number of parts > 1 and the number of partitions of n into an odd number of parts > 1.

Convolution inverse of A025147.

Index entries for related partition-counting sequences

Cf. A000700, A002865, A025147, A027349, A078616, A081362.

nmax = 78; CoefficientList[Series[Product[1/(1 + x^k), {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} 1/(1 + x^k).
a(n) = (-1)^n * (A000700(n) - A000700(n-1)), for n > 0. - Vaclav Kotesovec, Jun 06 2018
a(n) ~ (-1)^n * Pi * exp(Pi*sqrt(n/6)) / (2^(13/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Jun 06 2018
a(n) = A027188(n+2) - A027194(n+2). - R. J. Mathar, Jun 16 2018

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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A027194 Number of partitions of n into an even number of parts, the least being 2; also, a(n+2) = number of partitions of n into an odd number of parts, each >=2.

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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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A298596 Expansion of Product_{k>=2} 1/(1 + x^k).

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