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A022169 Triangle of Gaussian binomial coefficients [ n,k ] for q = 5.

%I A022169 #46 May 09 2025 08:35:22
%S A022169 1,1,1,1,6,1,1,31,31,1,1,156,806,156,1,1,781,20306,20306,781,1,1,3906,
%T A022169 508431,2558556,508431,3906,1,1,19531,12714681,320327931,320327931,
%U A022169 12714681,19531,1,1,97656,317886556
%N A022169 Triangle of Gaussian binomial coefficients [ n,k ] for q = 5.
%C A022169 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 A022169 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%D A022169 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A022169 G. C. Greubel, <a href="/A022169/b022169.txt">Rows n=0..50 of triangle, flattened</a>
%H A022169 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 A022169 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 A022169 <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries for sequences related to Gaussian binomial coefficients</a>
%F A022169 T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - _Peter A. Lawrence_, Jul 13 2017
%F A022169 G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 5^j - 1. - _Seiichi Manyama_, May 09 2025
%e A022169 Triangle begins:
%e A022169   1;
%e A022169   1,     1;
%e A022169   1,     6,        1;
%e A022169   1,    31,       31,         1;
%e A022169   1,   156,      806,       156,         1;
%e A022169   1,   781,    20306,     20306,       781,        1;
%e A022169   1,  3906,   508431,   2558556,    508431,     3906,     1;
%e A022169   1, 19531, 12714681, 320327931, 320327931, 12714681, 19531, 1,
%p A022169 A027872 := proc(n)
%p A022169         mul( 5^i-1, i=1..n) ;
%p A022169 end proc:
%p A022169 A022169 := proc(n, m)
%p A022169         A027872(n)/A027872(n-m)/A027872(m) ;
%p A022169 end proc: # _R. J. Mathar_, Mar 12 2013
%t A022169 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 A022169 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 A022169 (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 A022169 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018
%Y A022169 Cf. A003462 (column k=1), A006111 (k=2), A006112 (k=3).
%Y A022169 Row sums give A006119.
%K A022169 nonn,tabl
%O A022169 0,5
%A A022169 _N. J. A. Sloane_