A026921 Triangular array E by rows: E(n,k) = number of partitions of n into an even number of parts, the greatest being k.

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 2, 1, 2, 1, 1, 0, 1, 2, 3, 2, 2, 1, 1, 0, 0, 2, 3, 3, 2, 2, 1, 1, 0, 1, 2, 5, 4, 4, 2, 2, 1, 1, 0, 0, 3, 4, 6, 4, 4, 2, 2, 1, 1, 0, 1, 3, 7, 7, 7, 5, 4, 2, 2, 1, 1, 0, 0, 3, 6, 10, 8, 7, 5, 4, 2, 2, 1, 1, 0, 1, 3, 9, 11, 13, 9, 8, 5, 4, 2, 2, 1, 1, 0

Offset

1,18

Comments

The reversed rows (see example) stabilize to A027193. [Joerg Arndt, May 12 2013]

Links

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

Formula

G.f. (including term a(0)=1): sum(n>=0, q^(2*n)/prod(k=1..2*n, 1-z*q^k) ), set z=1 to obtain g.f. for A027187. [Joerg Arndt, May 12 2013]

A026920(n,k) + E(n,k) = A008284(n,k). - R. J. Mathar, Aug 23 2019

Example

G.f. = (1)*q^0 +
(0) * q^1 +
(1 + 0*z) * q^2 +
(0 + 1*z + 0*z^2) * q^3 +
(1 + 1*z + 1*z^2 + 0*z^3) * q^4 +
(0 + 1*z + 1*z^2 + 1*z^3 + 0*z^4) * q^5 +
(1 + 1*z + 2*z^2 + 1*z^3 + 1*z^4 + 0*z^5) * q^6 +
(0 + 2*z + 1*z^2 + 2*z^3 + 1*z^4 + 1*z^5 + 0*z^6) * q^7 +
... [Joerg Arndt, May 12 2013]
Triangle starts:
01: [0]
02: [1, 0]
03: [0, 1, 0]
04: [1, 1, 1, 0]
05: [0, 1, 1, 1, 0]
06: [1, 1, 2, 1, 1, 0]
07: [0, 2, 1, 2, 1, 1, 0]
08: [1, 2, 3, 2, 2, 1, 1, 0]
09: [0, 2, 3, 3, 2, 2, 1, 1, 0]
10: [1, 2, 5, 4, 4, 2, 2, 1, 1, 0]
11: [0, 3, 4, 6, 4, 4, 2, 2, 1, 1, 0]
12: [1, 3, 7, 7, 7, 5, 4, 2, 2, 1, 1, 0]
13: [0, 3, 6, 10, 8, 7, 5, 4, 2, 2, 1, 1, 0]
14: [1, 3, 9, 11, 13, 9, 8, 5, 4, 2, 2, 1, 1, 0]
15: [0, 4, 8, 14, 14, 14, 9, 8, 5, 4, 2, 2, 1, 1, 0]
16: [1, 4, 12, 16, 20, 17, 15, 10, 8, 5, 4, 2, 2, 1, 1, 0]
17: [0, 4, 11, 20, 22, 23, 18, 15, 10, 8, 5, 4, 2, 2, 1, 1, 0]
18: [1, 4, 15, 23, 30, 28, 26, 19, 16, 10, 8, 5, 4, 2, 2, 1, 1, 0]
19: [0, 5, 13, 28, 33, 37, 31, 27, 19, 16, 10, 8, 5, 4, 2, 2, 1, 1, 0]
20: [1, 5, 18, 31, 44, 44, 43, 34, 28, 20, 16, 10, 8, 5, 4, 2, 2, 1, 1, 0]
... [Joerg Arndt, May 12 2013]

Prog

(PARI)
N = 20;  q = 'q + O('q^N);
gf = sum(n=0, N, q^(2*n)/prod(k=1, 2*n, 1-'z*q^k) );
v = Vec(gf);
{ for(n=2, #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-2, j, p[j]);
    print(p);
); }
/* Joerg Arndt, May 12 2013 */

Crossrefs

E(n, k) = O(n-k, 1)+O(n-k, 2)+...+O(n-k, m), where m=MIN{k, n-k}, n >= 2, O given by A026920.

Columns k=3..6: A026927, A026928, A026929, A026930.

Sequence in context: A029412 A368571 A178670 * A198568 A327310 A232539

Adjacent sequences: A026918 A026919 A026920 * A026922 A026923 A026924

Keyword

nonn,tabl

Author

Clark Kimberling

Status

approved