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A173118 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 = 2, read by rows.

%I A173118 #8 Apr 28 2021 02:00:40
%S A173118 1,1,1,1,4,1,1,5,5,1,1,6,10,6,1,1,7,20,20,7,1,1,8,27,40,27,8,1,1,9,35,
%T A173118 75,75,35,9,1,1,10,44,110,150,110,44,10,1,1,11,54,154,276,276,154,54,
%U A173118 11,1,1,12,65,208,430,552,430,208,65,12,1
%N A173118 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 = 2, read by rows.
%H A173118 G. C. Greubel, <a href="/A173118/b173118.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A173118 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 = 2.
%F A173118 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 = 2. - _G. C. Greubel_, Apr 27 2021
%e A173118 Triangle begins as:
%e A173118   1;
%e A173118   1,  1;
%e A173118   1,  4,  1;
%e A173118   1,  5,  5,   1;
%e A173118   1,  6, 10,   6,   1;
%e A173118   1,  7, 20,  20,   7,   1;
%e A173118   1,  8, 27,  40,  27,   8,   1;
%e A173118   1,  9, 35,  75,  75,  35,   9,   1;
%e A173118   1, 10, 44, 110, 150, 110,  44,  10,  1;
%e A173118   1, 11, 54, 154, 276, 276, 154,  54, 11,  1;
%e A173118   1, 12, 65, 208, 430, 552, 430, 208, 65, 12, 1;
%t A173118 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 A173118 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 27 2021 *)
%o A173118 (Sage)
%o A173118 @CachedFunction
%o A173118 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 A173118 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 27 2021
%Y A173118 Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), this sequence (q=2), A173119 (q=3), A173120 (q= -4), A173122.
%K A173118 nonn,tabl,easy,less
%O A173118 0,5
%A A173118 _Roger L. Bagula_, Feb 10 2010
%E A173118 Edited by _G. C. Greubel_, Apr 27 2021