A026921 -id:A026921 - cp's OEIS frontend

A026927 Number of partitions of n into an even number of parts, the greatest being 3; also, a(n+5) = number of partitions of n+2 into an odd number of parts, each <= 3.

0, 0, 0, 1, 1, 2, 1, 3, 3, 5, 4, 7, 6, 9, 8, 12, 11, 15, 13, 18, 17, 22, 20, 26, 24, 30, 28, 35, 33, 40, 37, 45, 43, 51, 48, 57, 54, 63, 60, 70, 67, 77, 73, 84, 81, 92, 88, 100, 96, 108, 104, 117

Offset: 1

Clark Kimberling

nonn

			Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
                                                          1+1+8
                                                   1+1+7  1+2+7
                                                   1+2+6  1+3+6
                                            1+1+6  1+3+5  1+4+5
                                     1+1+5  1+2+5  1+4+4  2+2+6
                              1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
                       1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
         1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...
-----------------------------------------------------------------------
  n  |     3      4      5      6      7      8      9     10      ...
-----------------------------------------------------------------------
a(n) |     0      1      1      2      1      3      3      5      ...
-----------------------------------------------------------------------

3rd column of A026921.

Cf. A026923, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309692, A309694.

a(n) + A026923(n) = A069905(n). - R. J. Mathar, Aug 22 2019
Conjectures from Colin Barker, Sep 01 2019: (Start)
G.f.: x^4*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) + a(n-6) - a(n-7) - a(n-10) + a(n-11) for n>11.
(End)

A026920 Triangular array O by rows: O(n,k) = number of partitions of n into an odd number of parts, the greatest being k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 1, 3, 1, 1, 0, 1, 0, 2, 2, 3, 1, 1, 0, 1, 1, 2, 4, 3, 3, 1, 1, 0, 1, 0, 3, 3, 5, 3, 3, 1, 1, 0, 1, 1, 2, 6, 5, 6, 3, 3, 1, 1, 0, 1, 0, 3, 5, 8, 6, 6, 3, 3, 1, 1, 0, 1, 1, 3, 8, 8, 10, 7, 6, 3, 3, 1, 1, 0, 1, 0, 4, 7, 12, 10, 11, 7, 6, 3, 3, 1, 1, 0, 1

Offset: 1

Clark Kimberling

The reversed rows (see example) stabilize to A027187. [Joerg Arndt, May 12 2013]

			G.f. = (0)*q^0 +
(1) * q^1
(0* + 1*z^1) * q^2
(1* + 0*z^1 + 1*z^2) * q^3
(0* + 1*z^1 + 0*z^2 + 1*z^3) * q^4
(1* + 1*z^1 + 1*z^2 + 0*z^3 + 1*z^4) * q^5
(0* + 2*z^1 + 1*z^2 + 1*z^3 + 0*z^4 + 1*z^5) * q^6
(1* + 1*z^1 + 3*z^2 + 1*z^3 + 1*z^4 + 0*z^5 + 1*z^6) * q^7
... [_Joerg Arndt_, May 12 2013]
Triangle starts:
01: [1]
02: [0, 1]
03: [1, 0, 1]
04: [0, 1, 0, 1]
05: [1, 1, 1, 0, 1]
06: [0, 2, 1, 1, 0, 1]
07: [1, 1, 3, 1, 1, 0, 1]
08: [0, 2, 2, 3, 1, 1, 0, 1]
09: [1, 2, 4, 3, 3, 1, 1, 0, 1]
10: [0, 3, 3, 5, 3, 3, 1, 1, 0, 1]
11: [1, 2, 6, 5, 6, 3, 3, 1, 1, 0, 1]
12: [0, 3, 5, 8, 6, 6, 3, 3, 1, 1, 0, 1]
13: [1, 3, 8, 8, 10, 7, 6, 3, 3, 1, 1, 0, 1]
14: [0, 4, 7, 12, 10, 11, 7, 6, 3, 3, 1, 1, 0, 1]
15: [1, 3, 11, 13, 16, 12, 12, 7, 6, 3, 3, 1, 1, 0, 1]
16: [0, 4, 9, 18, 17, 18, 13, 12, 7, 6, 3, 3, 1, 1, 0, 1]
17: [1, 4, 13, 19, 25, 21, 20, 14, 12, 7, 6, 3, 3, 1, 1, 0, 1]
18: [0, 5, 12, 24, 27, 30, 23, 21, 14, 12, 7, 6, 3, 3, 1, 1, 0, 1]
19: [1, 4, 17, 26, 37, 34, 34, 25, 22, 14, 12, 7, 6, 3, 3, 1, 1, 0, 1]
... [_Joerg Arndt_, May 12 2013]

O(n, k) = E(n-k, 1)+E(n-k, 2)+...+E(n-k, m), where m=MIN{k, n-k}, n >= 2, E given by A026921.

Columns k=2..6: A026922, A026923, A026924, A026925, A026926.

N = 20;  q = 'q + O('q^N);
gf = sum(n=0,N, q^(2*n+1)/prod(k=1, 2*n+1, 1-'z*q^k) );
v = Vec(gf);
{ for(n=1, #v, /* print triangle starting with row 1: */
    p = Pol('c0 +'cn*'z^n + v[n],'z);
    p = polrecip(p);
    p = Vec(p);
    p[1] -= 'c0;
    p = vector(#p-1, j, p[j]);
    print(p);
); }
/* Joerg Arndt, May 12 2013 */

G.f.: sum(n>=0, q^(2*n+1)/prod(k=1..2*n+1, 1-z*q^k) ), setting z=1 gives g.f. for A027193. [Joerg Arndt, May 12 2013]

O(n,k) + A026921(n,k) = A008284(n,k). - R. J. Mathar, Aug 23 2019

A026928 Number of partitions of n into an even number of parts, the greatest being 4; also, a(n+7) = number of partitions of n+3 into an odd number of parts, each <=4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 6, 7, 10, 11, 14, 16, 20, 23, 28, 31, 37, 41, 48, 53, 61, 67, 76, 83, 94, 102, 114, 123, 136, 147, 162, 174, 191, 204, 222, 237, 257, 274, 296, 314, 338, 358, 384, 406, 434, 458, 488, 514

Offset: 1

Clark Kimberling

nonn

4th column of A026921.

A026929 Number of partitions of n into an even number of parts, the greatest being 5; also, a(n+9) = number of partitions of n+4 into an odd number of parts, each <=5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 14, 20, 22, 30, 33, 44, 48, 62, 68, 85, 93, 114, 124, 149, 162, 193, 209, 245, 265, 307, 331, 380, 409, 466, 500, 565, 605, 679, 726, 810, 864, 959, 1021, 1127, 1198, 1317, 1397, 1529, 1620, 1767

Offset: 1

Clark Kimberling

nonn

5th column of A026921.

A026930 Number of partitions of n into an even number of parts, the greatest being 6; also, a(n+11) = number of partitions of n+5 into an odd number of parts, each <=6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 7, 9, 14, 17, 23, 28, 37, 44, 56, 66, 83, 98, 119, 139, 168, 194, 229, 263, 309, 352, 408, 462, 532, 600, 683, 766, 869, 969, 1090, 1211, 1356, 1500, 1670, 1840, 2041, 2242, 2473, 2707, 2978, 3249

Offset: 1

Clark Kimberling

nonn

6th column of A026921.

Table[Count[IntegerPartitions[n],?(EvenQ[Length[#]]&&#[[1]]==6&)],{n,60}] (* _Harvey P. Dale, May 04 2025 *)

A026932 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, 1, 0, 1, 1, 1, 2, 3, 3, 0, 1, 2, 3, 3, 1, 2, 4, 5, 6, 6, 0, 2, 3, 5, 6, 7, 7, 1, 3, 6, 8, 10, 11, 12, 12, 0, 2, 5, 8, 10, 12, 13, 14, 14, 1, 3, 8, 12, 16, 18, 20, 21, 22, 22, 0, 3, 7, 13, 17, 21, 23, 25, 26, 27, 27, 1, 4, 11, 18, 25, 30, 34, 36, 38

Offset: 1

Clark Kimberling

Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 20.

Cf. A026921.

T(n, k) = Sum_{i=1..k} A026921(n, i).

cp's OEIS Frontend

A026927 Number of partitions of n into an even number of parts, the greatest being 3; also, a(n+5) = number of partitions of n+2 into an odd number of parts, each <= 3.

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A026920 Triangular array O by rows: O(n,k) = number of partitions of n into an odd number of parts, the greatest being k.

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A026928 Number of partitions of n into an even number of parts, the greatest being 4; also, a(n+7) = number of partitions of n+3 into an odd number of parts, each <=4.

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A026929 Number of partitions of n into an even number of parts, the greatest being 5; also, a(n+9) = number of partitions of n+4 into an odd number of parts, each <=5.

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A026930 Number of partitions of n into an even number of parts, the greatest being 6; also, a(n+11) = number of partitions of n+5 into an odd number of parts, each <=6.

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Mathematica

A026932 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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