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

%I A022171 #37 May 09 2025 08:35:02
%S A022171 1,1,1,1,8,1,1,57,57,1,1,400,2850,400,1,1,2801,140050,140050,2801,1,1,
%T A022171 19608,6865251,48177200,6865251,19608,1,1,137257,336416907,
%U A022171 16531644851,16531644851,336416907,137257,1,1
%N A022171 Triangle of Gaussian binomial coefficients [ n,k ] for q = 7.
%D A022171 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%H A022171 G. C. Greubel, <a href="/A022171/b022171.txt">Rows n=0..50 of triangle, flattened</a>
%H A022171 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 A022171 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.
%H A022171 <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries for sequences related to Gaussian binomial coefficients</a>
%F A022171 T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - _Peter A. Lawrence_, Jul 13 2017
%F A022171 G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 7^j - 1. - _Seiichi Manyama_, May 09 2025
%e A022171 Triangle begins:
%e A022171   1;
%e A022171   1,      1;
%e A022171   1,      8,         1;
%e A022171   1,     57,        57,           1;
%e A022171   1,    400,      2850,         400,           1;
%e A022171   1,   2801,    140050,      140050,        2801,         1;
%e A022171   1,  19608,   6865251,    48177200,     6865251,     19608,      1;
%e A022171   1, 137257, 336416907, 16531644851, 16531644851, 336416907, 137257, 1;
%p A022171 A027875 := proc(n)
%p A022171     mul(7^i-1,i=1..n) ;
%p A022171 end proc:
%p A022171 A022171 := proc(n,m)
%p A022171     A027875(n)/A027875(m)/A027875(n-m) ;
%p A022171 end proc: # _R. J. Mathar_, Jul 19 2017
%t A022171 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 A022171 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 A022171 (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 A022171 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018
%Y A022171 Cf. A023000 (k=1), A022231 (k=2)
%K A022171 nonn,tabl
%O A022171 0,5
%A A022171 _N. J. A. Sloane_