A022171 Triangle of Gaussian binomial coefficients [ n,k ] for q = 7.

1, 1, 1, 1, 8, 1, 1, 57, 57, 1, 1, 400, 2850, 400, 1, 1, 2801, 140050, 140050, 2801, 1, 1, 19608, 6865251, 48177200, 6865251, 19608, 1, 1, 137257, 336416907, 16531644851, 16531644851, 336416907, 137257, 1, 1

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: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.

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 8 1;
1 57 57 1;
1 400 2850 400 1;
1 2801 140050 140050 2801 1;
1 19608 6865251 48177200 6865251 19608 1;
1 137257 336416907 16531644851 16531644851 336416907 137257 1;

Maple

A027875 := proc(n)
    mul(7^i-1, i=1..n) ;
end proc:
A022171 := proc(n, m)
    A027875(n)/A027875(m)/A027875(n-m) ;
end proc: # R. J. Mathar, Jul 19 2017

Mathematica

p[n_]:=Product[7^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, 7], {n, 0, 10}, {k, 0, n}]//Flatten (* or *) q:= 7; 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=7; 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. A023000 (k=1), A022231 (k=2)

Sequence in context: A259465 A176227 A340560 * A203443 A176642 A172346

Adjacent sequences: A022168 A022169 A022170 * A022172 A022173 A022174

Keyword

nonn,tabl

Author

N. J. A. Sloane

Status

approved