A022170 Triangle of Gaussian binomial coefficients [ n,k ] for q = 6.

1, 1, 1, 1, 7, 1, 1, 43, 43, 1, 1, 259, 1591, 259, 1, 1, 1555, 57535, 57535, 1555, 1, 1, 9331, 2072815, 12485095, 2072815, 9331, 1, 1, 55987, 74630671, 2698853335, 2698853335, 74630671, 55987, 1, 1, 335923

Offset

0,5

References

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

Links

G. C. Greubel, 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.

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 7 1;
1 43 43 1;
1 259 1591 259 1;
1 1555 57535 57535 1555 1;
1 9331 2072815 12485095 2072815 9331 1;
1 55987 74630671 2698853335 2698853335 74630671 55987 1 ;

Maple

A027873 := proc(n)
    mul(6^i-1, i=1..n) ;
end procc:
A022170 := proc(n, m)
    A027873(n)/A027873(m)/A027873(n-m) ;
end proc: # R. J. Mathar, Jul 19 2017

Mathematica

p[n_]:= Product[6^i - 1, {i, 1, n}]; t[n_, k_]:= p[n]/(p[k]*p[n-k]); Table[t[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* Vincenzo Librandi, Aug 13 2016 *)
Table[QBinomial[n, k, 6], {n, 0, 10}, {k, 0, n}]//Flatten (* or *) q:= 6; 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=6; 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. A003463 (k=1), A022220 (k=2), A022221 (k=3).

Sequence in context: A173584 A166973 A157156 * A178658 A156602 A203389

Adjacent sequences: A022167 A022168 A022169 * A022171 A022172 A022173

Keyword

nonn,tabl

Author

N. J. A. Sloane

Status

approved