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

%I A022170 #34 May 09 2025 08:43:26
%S A022170 1,1,1,1,7,1,1,43,43,1,1,259,1591,259,1,1,1555,57535,57535,1555,1,1,
%T A022170 9331,2072815,12485095,2072815,9331,1,1,55987,74630671,2698853335,
%U A022170 2698853335,74630671,55987,1,1,335923
%N A022170 Triangle of Gaussian binomial coefficients [ n,k ] for q = 6.
%D A022170 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%H A022170 G. C. Greubel, <a href="/A022170/b022170.txt">Rows n=0..50 of triangle, flattened</a>
%H A022170 R. Mestrovic, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.
%H A022170 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 A022170 <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries for sequences related to Gaussian binomial coefficients</a>
%F A022170 T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - _Peter A. Lawrence_, Jul 13 2017
%F A022170 G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 6^j - 1. - _Seiichi Manyama_, May 09 2025
%e A022170 Triangle begins:
%e A022170   1;
%e A022170   1,     1;
%e A022170   1,     7,        1;
%e A022170   1,    43,       43,          1;
%e A022170   1,   259,     1591,        259,          1;
%e A022170   1,  1555,    57535,      57535,       1555,        1;
%e A022170   1,  9331,  2072815,   12485095,    2072815,     9331,     1;
%e A022170   1, 55987, 74630671, 2698853335, 2698853335, 74630671, 55987, 1 ;
%p A022170 A027873 := proc(n)
%p A022170     mul(6^i-1,i=1..n) ;
%p A022170 end procc:
%p A022170 A022170 := proc(n,m)
%p A022170     A027873(n)/A027873(m)/A027873(n-m) ;
%p A022170 end proc: # _R. J. Mathar_, Jul 19 2017
%t A022170 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 *)
%t A022170 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 *)
%o A022170 (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))))};
%o A022170 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018
%Y A022170 Cf. A003463 (k=1), A022220 (k=2), A022221 (k=3).
%K A022170 nonn,tabl
%O A022170 0,5
%A A022170 _N. J. A. Sloane_