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A022173 Triangle of Gaussian binomial coefficients [ n,k ] for q = 9.

%I A022173 #41 May 09 2025 10:34:25
%S A022173 1,1,1,1,10,1,1,91,91,1,1,820,7462,820,1,1,7381,605242,605242,7381,1,
%T A022173 1,66430,49031983,441826660,49031983,66430,1,1,597871,3971657053,
%U A022173 322140667123,322140667123,3971657053,597871,1
%N A022173 Triangle of Gaussian binomial coefficients [ n,k ] for q = 9.
%D A022173 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%H A022173 G. C. Greubel, <a href="/A022173/b022173.txt">Rows n=0..50 of triangle, flattened</a>
%H A022173 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 A022173 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 A022173 <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries for sequences related to Gaussian binomial coefficients</a>
%F A022173 T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - _Peter A. Lawrence_, Jul 13 2017
%F A022173 G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 9^j - 1. - _Seiichi Manyama_, May 09 2025
%e A022173 Triangle begins:
%e A022173   1;
%e A022173   1,      1;
%e A022173   1,     10,          1;
%e A022173   1,     91,         91,            1;
%e A022173   1,    820,       7462,          820,            1;
%e A022173   1,   7381,     605242,       605242,         7381,          1;
%e A022173   1,  66430,   49031983,    441826660,     49031983,      66430,      1;
%e A022173   1, 597871, 3971657053, 322140667123, 322140667123, 3971657053, 597871, 1;
%p A022173 A027877 := proc(n)
%p A022173     mul(9^i-1,i=1..n) ;
%p A022173 end proc:
%p A022173 A022173 := proc(n,m)
%p A022173     A027877(n)/A027877(m)/A027877(n-m) ;
%p A022173 end proc: # _R. J. Mathar_, Jul 19 2017
%t A022173 a027878[n_]:=Times@@ Table[9^i - 1, {i, n}]; T[n_, m_]:=a027878[n]/( a027878[m] a027878[n-m]); Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* _Indranil Ghosh_, Jul 20 2017, after Maple code *)
%t A022173 Table[QBinomial[n,k,9], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 9; 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 A022173 (Python)
%o A022173 from operator import mul
%o A022173 def a027878(n): return 1 if n==0 else reduce(mul, [9**i - 1 for i in range(1, n + 1)])
%o A022173 def T(n, m): return a027878(n)/(a027878(m)*a027878(n - m))
%o A022173 for n in range(11): print([T(n, m) for m in range(n + 1)]) # _Indranil Ghosh_, Jul 20 2017, after Maple code
%o A022173 (PARI) {q=9; 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 A022173 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018
%K A022173 nonn,tabl
%O A022173 0,5
%A A022173 _N. J. A. Sloane_