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A026921 Triangular array E by rows: E(n,k) = number of partitions of n into an even number of parts, the greatest being k. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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]

A0269210(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: A117452 A029412 A178670 * A198568 A327310 A232539

Adjacent sequences:  A026918 A026919 A026920 * A026922 A026923 A026924

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified April 7 04:20 EDT 2020. Contains 333292 sequences. (Running on oeis4.)