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Triangle of Gaussian binomial coefficients [ n,k ] for q = 10.
17

%I #30 Mar 12 2021 15:58:10

%S 1,1,1,1,11,1,1,111,111,1,1,1111,11211,1111,1,1,11111,1122211,1122211,

%T 11111,1,1,111111,112232211,1123333211,112232211,111111,1,1,1111111,

%U 11223332211,1123445443211,1123445443211

%N Triangle of Gaussian binomial coefficients [ n,k ] for q = 10.

%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="/A022174/b022174.txt">Rows n=0..50 of triangle, flattened</a>

%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 11 1;

%e 1 111 111 1;

%e 1 1111 11211 1111 1;

%e 1 11111 1122211 1122211 11111 1;

%e 1 111111 112232211 1123333211 112232211 111111 1;

%e 1 1111111 11223332211 1123445443211 1123445443211 11223332211 1111111 1;

%p A027878 := proc(n)

%p mul(10^i-1,i=1..n) ;

%p end proc:

%p A022174 := proc(n,m)

%p A027878(n)/A027878(m)/A027878(n-m) ;

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

%t a027878[n_]:=Times@@ Table[10^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,10], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 10; 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 28 2018 *)

%o (Python)

%o from operator import mul

%o def a027878(n): return 1 if n==0 else reduce(mul, [10**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=10; 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 28 2018

%Y Row sums give A015196.

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_