0,2

There are A033638(n) values in the n-th row, compliant with the order of the polynomial.

In the example for n=6 detailed below, the orders of [6,k]_q are 1,6,9,10,9,6,1 for k=0..6,

the maximum order 10 defining the row length

Note also that 1 6 9 10 9 6 1 and related distributions are antidiagonals of A077028.

A083480 is a variation illustrating a relationship with numeric partitions, A000041.

The rows are formed by the nonzero entries of the columns of A049597.

Andrews(1976) Theory of Partitions (page 242)

Eric Weisstein, q-Binomial Coefficient, Mathworld.

Row sums: sum_k T(n,k)= 2^n.

When viewed as an array with A033638(r) entries per row, the table begins:

. 1 ....... : 1

. 2 ....... : 2

. 3 1 ....... : 3+q = (1)+(1+q)+(1)

. 4 2 2 ....... : 4+2q+2q^2 = 1+(1+q+q^2)+(1+q+q^2)+1

. 5 3 4 3 1 ....... : 5+3q+4q^2+3q^3+q^4

. 6 4 6 6 6 2 2

. 7 5 8 9 11 9 7 4 3 1 ....... : 7+5q+8q^2+9q^3+11q^4+9q^5+...

. 8 6 10 12 16 16 18 12 12 8 6 2 2

. 9 7 12 15 21 23 29 27 26 23 21 15 13 7 4 3 1

...

This last row is from the sum over 7 q-polynomials coefficients ....... :

. 1 ....... : 1 = [6,0]_q

. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,1]_q

. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,2]_q

. 1 1 2 3 3 3 3 2 1 1 ....... : 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9 = [6,3]_q

. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,4]_q

. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,5]_q

. 1 ....... : 1 = [6,6]_q

QBinomial := proc(n, m, q) local i ; factor( mul((1-q^(n-i))/(1-q^(i+1)), i=0..m-1) ) ; expand(%) ; end:

A083906 := proc(n, k) add( QBinomial(n, m, q), m=0..n ) ; coeftayl(%, q=0, k) ; end:

A033638 := proc(n) coeftayl( (1-x+x^3)/(1-x)^2/(1-x^2), x=0, n) ; end:

for n from 0 to 10 do for k from 0 to A033638(n)-1 do printf("%d, ", A083906(n, k)) ; od: od: # R. J. Mathar, May 28 2009

Cf. A033638, A077028, A083479, A083480.

Sequence in context: A104705 A143361 A152547 this_sequence A160541 A022446 A122196

Adjacent sequences: A083903 A083904 A083905 this_sequence A083907 A083908 A083909

nonn,tabf

Alford Arnold (Alford1940(AT)aol.com), Jun 19 2003

Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 28 2009