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A022168 Triangle of Gaussian binomial coefficients [ n,k ] for q = 4. 22
1, 1, 1, 1, 5, 1, 1, 21, 21, 1, 1, 85, 357, 85, 1, 1, 341, 5797, 5797, 341, 1, 1, 1365, 93093, 376805, 93093, 1365, 1, 1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1, 1, 21845, 23859109, 1550842085, 6221613541 (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*A157784(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.

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, 5, 1;

1, 21, 21, 1;

1, 85, 357, 85, 1;

1, 341, 5797, 5797, 341, 1;

1, 1365, 93093, 376805, 93093, 1365, 1;

1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1;

MAPLE

A022168 := proc(n, m)

        A027637(n)/A027637(n-m)/A027637(m) ;

end proc: # R. J. Mathar, Nov 14 2011

MATHEMATICA

gaussianBinom[n_, k_, q_] := Product[q^i - 1, {i, n}]/Product[q^j - 1, {j, n - k}]/Product[q^l - 1, {l, k}]; Column[Table[gaussianBinom[n, k, 4], {n, 0, 8}, {k, 0, n}], Center] (* Alonso del Arte, Nov 14 2011 *)

Table[QBinomial[n, k, 4], {n, 0, 10}, {k, 0, n}]//Flatten (* or *) q:= 4; T[n_, 0]:= 1; T[n_, n_]:= 1; T[n_, k_]:= T[n, k] = If[k < 0 || n < k, 0, T[n-1, k -1] +q^k*T[n-1, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten  (* G. C. Greubel, May 27 2018 *)

PROG

(PARI) {q=4; T(n, k) = if(k==0, 1, if (k==n, 1, if (k<0 || n<k, 0, T(n-1, k-1) + q^k*T(n-1, k))))};

for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 27 2018

CROSSREFS

Cf. A006118 (row sums), A002450 (k=1), A006105 (k=2), A006106 (k=3).

Sequence in context: A036969 A080249 A157154 * A157212 A156600 A329118

Adjacent sequences:  A022165 A022166 A022167 * A022169 A022170 A022171

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 9 19:51 EST 2019. Contains 329879 sequences. (Running on oeis4.)