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A022172 Triangle of Gaussian binomial coefficients [ n,k ] for q = 8.

%I A022172 #42 May 09 2025 10:44:49
%S A022172 1,1,1,1,9,1,1,73,73,1,1,585,4745,585,1,1,4681,304265,304265,4681,1,1,
%T A022172 37449,19477641,156087945,19477641,37449,1,1,299593,1246606473,
%U A022172 79936505481,79936505481,1246606473,299593,1
%N A022172 Triangle of Gaussian binomial coefficients [ n,k ] for q = 8.
%D A022172 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%H A022172 G. C. Greubel, <a href="/A022172/b022172.txt">Rows n=0..50 of triangle, flattened</a>
%H A022172 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 A022172 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 A022172 <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries for sequences related to Gaussian binomial coefficients</a>
%F A022172 T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - _Peter A. Lawrence_, Jul 13 2017
%F A022172 G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 8^j - 1. - _Seiichi Manyama_, May 09 2025
%e A022172 Triangle begins:
%e A022172   1;
%e A022172   1,      1;
%e A022172   1,      9,          1;
%e A022172   1,     73,         73,           1;
%e A022172   1,    585,       4745,         585,           1;
%e A022172   1,   4681,     304265,      304265,        4681,          1;
%e A022172   1,  37449,   19477641,   156087945,    19477641,      37449,      1;
%e A022172   1, 299593, 1246606473, 79936505481, 79936505481, 1246606473, 299593, 1;
%p A022172 A027876 := proc(n)
%p A022172     mul(8^i-1,i=1..n) ;
%p A022172 end proc:
%p A022172 A022172 := proc(n,m)
%p A022172     A027876(n)/A027876(m)/A027876(n-m) ;
%p A022172 end proc: # _R. J. Mathar_, Jul 19 2017
%t A022172 a027878[n_]:=Times@@ Table[8^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 *)
%t A022172 Table[QBinomial[n,k,8], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 8; 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 A022172 (Python)
%o A022172 from operator import mul
%o A022172 def a027878(n): return 1 if n==0 else reduce(mul, [8**i - 1 for i in range(1, n + 1)])
%o A022172 def T(n, m): return a027878(n)//(a027878(m)*a027878(n - m))
%o A022172 for n in range(11): print([T(n, m) for m in range(n + 1)]) # _Indranil Ghosh_, Jul 20 2017
%o A022172 (PARI) {q=8; 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 A022172 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018
%Y A022172 Cf. A023001 (k=1), A022242 (k=2).
%K A022172 nonn,tabl
%O A022172 0,5
%A A022172 _N. J. A. Sloane_