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

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

1,17

Comments

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

Links

Table of n, a(n) for n=1..105.

Formula

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

Example

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]

Prog

(PARI)
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 */

Crossrefs

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.

Sequence in context: A206588 A302234 A345007 * A060763 A131576 A341675

Adjacent sequences: A026917 A026918 A026919 * A026921 A026922 A026923

Keyword

nonn,tabl

Author

Clark Kimberling

Status

approved