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A022168 Triangle of Gaussian binomial coefficients [ n,k ] for q = 4.

%I A022168 #49 May 09 2025 08:33:23
%S A022168 1,1,1,1,5,1,1,21,21,1,1,85,357,85,1,1,341,5797,5797,341,1,1,1365,
%T A022168 93093,376805,93093,1365,1,1,5461,1490853,24208613,24208613,1490853,
%U A022168 5461,1,1,21845,23859109,1550842085,6221613541
%N A022168 Triangle of Gaussian binomial coefficients [ n,k ] for q = 4.
%C A022168 The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157784(n,k). - _R. J. Mathar_, Mar 12 2013
%D A022168 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%H A022168 T. D. Noe, <a href="/A022168/b022168.txt">Rows n=0..50 of triangle, flattened</a>
%H A022168 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 A022168 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 A022168 M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%H A022168 <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries for sequences related to Gaussian binomial coefficients</a>
%F A022168 T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - _Peter A. Lawrence_, Jul 13 2017
%F A022168 G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 4^j - 1. - _Seiichi Manyama_, May 09 2025
%e A022168 Triangle begins:
%e A022168   1;
%e A022168   1,    1;
%e A022168   1,    5,       1;
%e A022168   1,   21,      21,        1;
%e A022168   1,   85,     357,       85,        1;
%e A022168   1,  341,    5797,     5797,      341,       1;
%e A022168   1, 1365,   93093,   376805,    93093,    1365,    1;
%e A022168   1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1;
%p A022168 A022168 := proc(n,m)
%p A022168         A027637(n)/A027637(n-m)/A027637(m) ;
%p A022168 end proc: # _R. J. Mathar_, Nov 14 2011
%t A022168 gaussianBinom[n_, k_, q_] := Product[q^i - 1, {i, n}]/Product[q^j - 1, {j, n - k}]/Product[q^l - 1, {l, k}]; Column[Table[gaussianBinom[n, k, 4], {n, 0, 8}, {k, 0, n}], Center] (* _Alonso del Arte_, Nov 14 2011 *)
%t A022168 Table[QBinomial[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 4; 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 A022168 (PARI) {q=4; 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 A022168 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018
%Y A022168 Cf. A006118 (row sums), A002450 (k=1), A006105 (k=2), A006106 (k=3).
%K A022168 nonn,tabl
%O A022168 0,5
%A A022168 _N. J. A. Sloane_