Search: id:A083906
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%I A083906
%S A083906 1,2,3,1,4,2,2,5,3,4,3,1,6,4,6,6,6,2,2,7,5,8,9,11,9,7,4,3,1,8,6,10,12,
%T A083906 16,16,18,12,12,8,6,2,2,9,7,12,15,21,23,29,27,26,23,21,15,13,7,4,3,1,10,
%U A083906 8,14,18,26,30,40,42,48,44,46,40
%N A083906 Table T(n,k) read along rows: the coefficient [q^k] of the sum_{m=0..n} 
               [n,m]_q over q-Binomial coefficients.
%C A083906 There are A033638(n) values in the n-th row, compliant with the order 
               of the polynomial.
%C A083906 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,
%C A083906 the maximum order 10 defining the row length
%C A083906 Note also that 1 6 9 10 9 6 1 and related distributions are antidiagonals 
               of A077028.
%C A083906 A083480 is a variation illustrating a relationship with numeric partitions, 
               A000041.
%C A083906 The rows are formed by the nonzero entries of the columns of A049597.
%D A083906 Andrews(1976) Theory of Partitions (page 242)
%H A083906 Eric Weisstein, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">
               q-Binomial Coefficient</a>, Mathworld.
%F A083906 Row sums: sum_k T(n,k)= 2^n.
%e A083906 When viewed as an array with A033638(r) entries per row, the table begins:
%e A083906 . 1 ....... : 1
%e A083906 . 2 ....... : 2
%e A083906 . 3 1 ....... : 3+q = (1)+(1+q)+(1)
%e A083906 . 4 2 2 ....... : 4+2q+2q^2 = 1+(1+q+q^2)+(1+q+q^2)+1
%e A083906 . 5 3 4 3 1 ....... : 5+3q+4q^2+3q^3+q^4
%e A083906 . 6 4 6 6 6 2 2
%e A083906 . 7 5 8 9 11 9 7 4 3 1 ....... : 7+5q+8q^2+9q^3+11q^4+9q^5+...
%e A083906 . 8 6 10 12 16 16 18 12 12 8 6 2 2
%e A083906 . 9 7 12 15 21 23 29 27 26 23 21 15 13 7 4 3 1
%e A083906 ...
%e A083906 This last row is from the sum over 7 q-polynomials coefficients ....... 
               :
%e A083906 . 1 ....... : 1 = [6,0]_q
%e A083906 . 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,1]_q
%e A083906 . 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
%e A083906 . 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
%e A083906 . 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
%e A083906 . 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,5]_q
%e A083906 . 1 ....... : 1 = [6,6]_q
%p A083906 QBinomial := proc(n,m,q) local i ; factor( mul((1-q^(n-i))/(1-q^(i+1)),
               i=0..m-1) ) ; expand(%) ; end:
%p A083906 A083906 := proc(n,k) add( QBinomial(n,m,q),m=0..n ) ; coeftayl(%,q=0,
               k) ; end:
%p A083906 A033638 := proc(n) coeftayl( (1-x+x^3)/(1-x)^2/(1-x^2),x=0,n) ; end:
%p A083906 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
%Y A083906 Cf. A033638, A077028, A083479, A083480.
%Y A083906 Sequence in context: A104705 A143361 A152547 this_sequence A160541 A022446 
               A122196
%Y A083906 Adjacent sequences: A083903 A083904 A083905 this_sequence A083907 A083908 
               A083909
%K A083906 nonn,tabf
%O A083906 0,2
%A A083906 Alford Arnold (Alford1940(AT)aol.com), Jun 19 2003
%E A083906 Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 28 2009

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