A026920 - OEIS

%I #21 Aug 23 2019 04:51:13

%S 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,

%T 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,

%U 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

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

%C The reversed rows (see example) stabilize to A027187. [_Joerg Arndt_, May 12 2013]

%F 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]

%F O(n,k) + A026921(n,k) = A008284(n,k). - _R. J. Mathar_, Aug 23 2019

%e G.f. = (0)*q^0 +

%e (1) * q^1

%e (0* + 1*z^1) * q^2

%e (1* + 0*z^1 + 1*z^2) * q^3

%e (0* + 1*z^1 + 0*z^2 + 1*z^3) * q^4

%e (1* + 1*z^1 + 1*z^2 + 0*z^3 + 1*z^4) * q^5

%e (0* + 2*z^1 + 1*z^2 + 1*z^3 + 0*z^4 + 1*z^5) * q^6

%e (1* + 1*z^1 + 3*z^2 + 1*z^3 + 1*z^4 + 0*z^5 + 1*z^6) * q^7

%e ... [_Joerg Arndt_, May 12 2013]

%e Triangle starts:

%e 01: [1]

%e 02: [0, 1]

%e 03: [1, 0, 1]

%e 04: [0, 1, 0, 1]

%e 05: [1, 1, 1, 0, 1]

%e 06: [0, 2, 1, 1, 0, 1]

%e 07: [1, 1, 3, 1, 1, 0, 1]

%e 08: [0, 2, 2, 3, 1, 1, 0, 1]

%e 09: [1, 2, 4, 3, 3, 1, 1, 0, 1]

%e 10: [0, 3, 3, 5, 3, 3, 1, 1, 0, 1]

%e 11: [1, 2, 6, 5, 6, 3, 3, 1, 1, 0, 1]

%e 12: [0, 3, 5, 8, 6, 6, 3, 3, 1, 1, 0, 1]

%e 13: [1, 3, 8, 8, 10, 7, 6, 3, 3, 1, 1, 0, 1]

%e 14: [0, 4, 7, 12, 10, 11, 7, 6, 3, 3, 1, 1, 0, 1]

%e 15: [1, 3, 11, 13, 16, 12, 12, 7, 6, 3, 3, 1, 1, 0, 1]

%e 16: [0, 4, 9, 18, 17, 18, 13, 12, 7, 6, 3, 3, 1, 1, 0, 1]

%e 17: [1, 4, 13, 19, 25, 21, 20, 14, 12, 7, 6, 3, 3, 1, 1, 0, 1]

%e 18: [0, 5, 12, 24, 27, 30, 23, 21, 14, 12, 7, 6, 3, 3, 1, 1, 0, 1]

%e 19: [1, 4, 17, 26, 37, 34, 34, 25, 22, 14, 12, 7, 6, 3, 3, 1, 1, 0, 1]

%e ... [_Joerg Arndt_, May 12 2013]

%o (PARI)

%o N = 20;  q = 'q + O('q^N);

%o gf = sum(n=0,N, q^(2*n+1)/prod(k=1, 2*n+1, 1-'z*q^k) );

%o v = Vec(gf);

%o { for(n=1, #v, /* print triangle starting with row 1: */

%o     p = Pol('c0 +'cn*'z^n + v[n],'z);

%o     p = polrecip(p);

%o     p = Vec(p);

%o     p[1] -= 'c0;

%o     p = vector(#p-1, j, p[j]);

%o     print(p);

%o ); }

%o /* _Joerg Arndt_, May 12 2013 */

%Y 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.

%Y Columns k=2..6: A026922, A026923, A026924, A026925, A026926.

%K nonn,tabl

%O 1,17

%A _Clark Kimberling_