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%I #40 May 13 2019 15:25:28
%S 1,1,1,1,6,1,1,31,31,1,1,156,806,156,1,1,781,20306,20306,781,1,1,3906,
%T 508431,2558556,508431,3906,1,1,19531,12714681,320327931,320327931,
%U 12714681,19531,1,1,97656,317886556
%N Triangle of Gaussian binomial coefficients [ n,k ] for q = 5.
%C The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157832(n,k). - _R. J. Mathar_, Mar 12 2013
%D F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%D M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H G. C. Greubel, <a href="/A022169/b022169.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 preprint 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.
%H <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries for sequences related to Gaussian binomial coefficients</a>
%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, 6, 1;
%e 1, 31, 31, 1;
%e 1, 156, 806, 156, 1;
%e 1, 781, 20306, 20306, 781, 1;
%e 1, 3906, 508431, 2558556, 508431, 3906, 1;
%e 1, 19531, 12714681, 320327931, 320327931, 12714681, 19531, 1,
%p A027872 := proc(n)
%p mul( 5^i-1, i=1..n) ;
%p end proc:
%p A022169 := proc(n, m)
%p A027872(n)/A027872(n-m)/A027872(m) ;
%p end proc: # _R. J. Mathar_, Mar 12 2013
%t p[n_] := Product[5^i-1, {i, 1, n}]; t[n_, k_] := p[n]/(p[k]*p[n-k]); Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 14 2014 *)
%t Table[QBinomial[n,k,5], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 5; 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=5; 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. A003462 (column k=1), A006111 (k=2), A006112 (k=3).
%Y Row sums give A006119.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_
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