OFFSET
1,17
COMMENTS
The reversed rows (see example) stabilize to A027187. [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]
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
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved