login
Triangle of Gaussian binomial coefficients [ n,k ] for q = 9.
20

%I #30 Mar 12 2021 15:58:49

%S 1,1,1,1,10,1,1,91,91,1,1,820,7462,820,1,1,7381,605242,605242,7381,1,

%T 1,66430,49031983,441826660,49031983,66430,1,1,597871,3971657053,

%U 322140667123,322140667123,3971657053

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

%D F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.

%H G. C. Greubel, <a href="/A022173/b022173.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.

%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 10 1;

%e 1 91 91 1;

%e 1 820 7462 820 1;

%e 1 7381 605242 605242 7381 1;

%e 1 66430 49031983 441826660 49031983 66430 1;

%e 1 597871 3971657053 322140667123 322140667123 3971657053 597871 1;

%e 1 5380840 321704819164 234844517989720 2113887057661126 234844517989720 321704819164 5380840 1 ;

%p A027877 := proc(n)

%p mul(9^i-1,i=1..n) ;

%p end proc:

%p A022173 := proc(n,m)

%p A027877(n)/A027877(m)/A027877(n-m) ;

%p end proc: # _R. J. Mathar_, Jul 19 2017

%t 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 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 (Python)

%o from operator import mul

%o def a027878(n): return 1 if n==0 else reduce(mul, [9**i - 1 for i in range(1, n + 1)])

%o def T(n, m): return a027878(n)/(a027878(m)*a027878(n - m))

%o for n in range(11): print([T(n, m) for m in range(n + 1)]) # _Indranil Ghosh_, Jul 20 2017, after Maple code

%o (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 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 27 2018

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_