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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. 10

%I

%S 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 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,

%U 10,8,14,18,26,30,40,42,48,44,46,40

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

%C 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 the maximum order 10 defining the row length.

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

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

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

%C The coefficient of q^j in the Gaussian polynomial [n,m]_q is the number of binary words on alphabet {0,1} of length n having m 1's and j inversions. Hence T(n,k) is the number of length n binary words with exactly k inversions. - _Geoffrey Critzer_, May 14 2017

%C If n is even the n-th row converges to n+1, n-1, n-4, ..., 19, 13, 7, 4, 3, 1 which is A029552 reversed, and if n is odd the sequence is twice A098613. - _Michael Somos_, Jun 25 2017

%D Andrews(1976) Theory of Partitions (page 242)

%H Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.

%H Alexander Gruber, <a href="http://math.stackexchange.com/q/206890/">"The Egg:" Bizarre behavior of the roots of a family of polynomials</a> Mathematics StackExchange Oct 04 2012

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>, Mathworld.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Q-binomial">q-binomial</a>

%H <a href="/index/Coa#codes_binary_linear">Index entries for sequences related to binary linear codes</a>

%F Row sums: Sum_k T(n,k) = 2^n.

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

%e . 1 ....... : 1

%e . 2 ....... : 2

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

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

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

%e . 6 4 6 6 6 2 2

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

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

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

%e ...

%e The second but last row is from the sum over 7 q-polynomials coefficients ....... :

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

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

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

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

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

%p 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

%t Table[CoefficientList[Total[Table[FunctionExpand[QBinomial[n, k, q]], {k, 0, n}]],q], {n, 0, 10}] // Grid (* _Geoffrey Critzer_, May 14 2017 *)

%o (PARI) {T(n, k) = polcoeff(sum(m=0, n, prod(k=0, m-1, (x^n - x^k) / (x^m - x^k))), k)}; /* _Michael Somos_, Jun 25 2017 */

%Y Cf. A029552, A033638, A077028, A083479, A083480, A098613.

%K nonn,tabf

%O 0,2

%A _Alford Arnold_, Jun 19 2003

%E Edited by _R. J. Mathar_, May 28 2009

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Last modified April 18 22:08 EDT 2019. Contains 322237 sequences. (Running on oeis4.)