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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
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, 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, 8, 14, 18, 26, 30, 40, 42, 48, 44, 46, 40 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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

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

REFERENCES

Andrews(1976) Theory of Partitions (page 242)

LINKS

Table of n, a(n) for n=0..70.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Alexander Gruber, "The Egg:" Bizarre behavior of the roots of a family of polynomials Mathematics StackExchange Oct 04 2012

Eric Weisstein, q-Binomial Coefficient, Mathworld.

Wikipedia, q-binomial

Index entries for sequences related to binary linear codes

FORMULA

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

EXAMPLE

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

...

The second but 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

MAPLE

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:

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

MATHEMATICA

Table[CoefficientList[Total[Table[FunctionExpand[QBinomial[n, k, q]], {k, 0, n}]], q], {n, 0, 10}] // Grid (* Geoffrey Critzer, May 14 2017 *)

PROG

(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 */

CROSSREFS

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

Sequence in context: A104705 A143361 A152547 * A160541 A286001 A304106

Adjacent sequences:  A083903 A083904 A083905 * A083907 A083908 A083909

KEYWORD

nonn,tabf

AUTHOR

Alford Arnold, Jun 19 2003

EXTENSIONS

Edited by R. J. Mathar, May 28 2009

STATUS

approved

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Last modified March 25 12:11 EDT 2019. Contains 321470 sequences. (Running on oeis4.)