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A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3. 22
1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 130, 40, 1, 1, 121, 1210, 1210, 121, 1, 1, 364, 11011, 33880, 11011, 364, 1, 1, 1093, 99463, 925771, 925771, 99463, 1093, 1, 1, 3280, 896260, 25095280, 75913222, 25095280 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157783(n,k). - R. J. Mathar, Mar 12 2013

REFERENCES

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.

M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

LINKS

T. D. Noe, Rows n=0..50 of triangle, flattened

R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Index entries for sequences related to Gaussian binomial coefficients

FORMULA

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017

EXAMPLE

1;

1, 1;

1, 4, 1;

1, 13, 13, 1;

1, 40, 130, 40, 1;

1, 121, 1210, 1210, 121, 1;

1, 364, 11011, 33880, 11011, 364, 1;

1, 1093, 99463, 925771, 925771, 99463, 1093, 1;

1, 3280, 896260, 25095280, 75913222, 25095280,..

MAPLE

A022167 := proc(n, m)

        A027871(n)/A027871(n-m)/A027871(m) ;

end proc:

seq(seq(A022167(n, m), m=0..n), n=0..10) ; # R. J. Mathar, Nov 14 2011

MATHEMATICA

p[n_] := Product[3^k-1, {k, 1, n}]; t[n_, m_] := p[n]/(p[n-m]*p[m]); Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jan 10 2014, after R. J. Mathar *)

CROSSREFS

Cf. A006117 (row sums), A003462 (column k=1), A006100 (k=2), A006101 (k=3).

Sequence in context: A157153 A212801 A147565 * A064281 A267318 A050154

Adjacent sequences:  A022164 A022165 A022166 * A022168 A022169 A022170

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 15 11:43 EST 2019. Contains 329999 sequences. (Running on oeis4.)