A027194 -id:A027194 - cp's OEIS frontend

A027188 a(n) = number of partitions of n into an odd number of parts, the least being 2; also a(n+2) = number of partitions of n into an even number of parts, each >=2.

0, 0, 1, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 11, 12, 17, 20, 28, 33, 44, 52, 69, 82, 105, 126, 161, 191, 239, 286, 355, 423, 520, 618, 755, 896, 1084, 1285, 1549, 1829, 2190, 2583, 3079, 3621, 4297, 5041, 5960, 6977, 8214, 9595, 11264, 13123, 15353, 17854, 20828

Offset: 0

Clark Kimberling

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<2, 0,
       b(n, i-1, t) +`if`(i>n, 0, b(n-i, i, 1-t))))
    end:
a:= n-> b(n-2$2, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 27 2014

b[n_, i_, t_] := b[n, i, t] = If[n==0, t, If[i<2, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, 1-t]]]]; a[n_] := b[n-2, n-2, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 08 2015, after Alois P. Heinz *)

gf=sum(n=0, N,  q^(2*n+1)/prod(k=1,n, 1-q^(2*k)) * q^(2*n+1)/prod(k=1,n, 1-q^(2*k-1)) );
concat([0], Vec(gf) ) \\ Joerg Arndt, Feb 27 2014

G.f.: sum(n>=0,  q^(2*n+1)/prod(k=1..n, 1-q^(2*k)) * q^(2*n+1)/prod(k=1..n, 1-q^(2*k-1)) ). [Joerg Arndt, Feb 27 2014]
a(n+2) + A027194(n+2) = A002865(n). - R. J. Mathar, Jun 16 2018
G.f.: x^2 * Sum_{k>=0} x^(4*k)/Product_{j=1..2*k} (1-x^j). - Seiichi Manyama, May 15 2023
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3*2^(7/2)*n^(3/2)). - Vaclav Kotesovec, Jun 20 2025

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

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

A027188 a(n) = number of partitions of n into an odd number of parts, the least being 2; also a(n+2) = number of partitions of n into an even number of parts, each >=2.

Views

Author

Keywords

Links

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula