A022171 - OEIS

%I #25 May 28 2018 11:13:51

%S 1,1,1,1,8,1,1,57,57,1,1,400,2850,400,1,1,2801,140050,140050,2801,1,1,

%T 19608,6865251,48177200,6865251,19608,1,1,137257,336416907,

%U 16531644851,16531644851,336416907,137257,1,1

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

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

%H G. C. Greubel, <a href="/A022171/b022171.txt">Rows n=0..50 of triangle, flattened</a>

%H R. Mestrovic, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>,  arXiv:1409.3820 [math.NT], 2014.

%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

%F T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - _Peter A. Lawrence_, Jul 13 2017

%e 1 ;

%e 1 1;

%e 1 8 1;

%e 1 57 57 1;

%e 1 400 2850 400 1;

%e 1 2801 140050 140050 2801 1;

%e 1 19608 6865251 48177200 6865251 19608 1;

%e 1 137257 336416907 16531644851 16531644851 336416907 137257 1;

%p A027875 := proc(n)

%p     mul(7^i-1,i=1..n) ;

%p end proc:

%p A022171 := proc(n,m)

%p     A027875(n)/A027875(m)/A027875(n-m) ;

%p end proc: # _R. J. Mathar_, Jul 19 2017

%t 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 *)

%t 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 *)

%o (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))))};

%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018

%Y Cf. A023000 (k=1), A022231 (k=2)

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_