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

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

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

Examples

			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]
		

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.

Programs

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 */

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