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A173119 Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 3, read by rows.

%I A173119 #8 Apr 28 2021 01:59:59
%S A173119 1,1,1,1,5,1,1,6,6,1,1,7,12,7,1,1,8,28,28,8,1,1,9,36,56,36,9,1,1,10,
%T A173119 45,119,119,45,10,1,1,11,55,164,238,164,55,11,1,1,12,66,219,483,483,
%U A173119 219,66,12,1,1,13,78,285,702,966,702,285,78,13,1
%N A173119 Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 3, read by rows.
%H A173119 G. C. Greubel, <a href="/A173119/b173119.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A173119 T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 3.
%F A173119 Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = 3. - _G. C. Greubel_, Apr 27 2021
%e A173119 Triangle begins as:
%e A173119   1;
%e A173119   1,  1;
%e A173119   1,  5,  1;
%e A173119   1,  6,  6,   1;
%e A173119   1,  7, 12,   7,   1;
%e A173119   1,  8, 28,  28,   8,   1;
%e A173119   1,  9, 36,  56,  36,   9,   1;
%e A173119   1, 10, 45, 119, 119,  45,  10,   1;
%e A173119   1, 11, 55, 164, 238, 164,  55,  11,  1;
%e A173119   1, 12, 66, 219, 483, 483, 219,  66, 12,  1;
%e A173119   1, 13, 78, 285, 702, 966, 702, 285, 78, 13, 1;
%t A173119 T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j,0,5}]];
%t A173119 Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 27 2021 *)
%o A173119 (Sage)
%o A173119 @CachedFunction
%o A173119 def T(n,k,q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
%o A173119 flatten([[T(n,k,3) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Apr 27 2021
%Y A173119 Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), A173118 (q=2), this sequence (q=3), A173120 (q= -4), A173122.
%K A173119 nonn,tabl,easy,less
%O A173119 0,5
%A A173119 _Roger L. Bagula_, Feb 10 2010
%E A173119 Edited by _G. C. Greubel_, Apr 27 2021