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A173117 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 = 1, read by rows.

%I A173117 #8 Apr 28 2021 01:59:15
%S A173117 1,1,1,1,3,1,1,4,4,1,1,5,8,5,1,1,6,14,14,6,1,1,7,20,28,20,7,1,1,8,27,
%T A173117 49,49,27,8,1,1,9,35,76,98,76,35,9,1,1,10,44,111,175,175,111,44,10,1,
%U A173117 1,11,54,155,286,350,286,155,54,11,1
%N A173117 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 = 1, read by rows.
%H A173117 G. C. Greubel, <a href="/A173117/b173117.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A173117 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 = 1.
%F A173117 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 = 1. - _G. C. Greubel_, Apr 27 2021
%e A173117 Triangle begins as:
%e A173117   1;
%e A173117   1,  1;
%e A173117   1,  3,  1;
%e A173117   1,  4,  4,   1;
%e A173117   1,  5,  8,   5,   1;
%e A173117   1,  6, 14,  14,   6,   1;
%e A173117   1,  7, 20,  28,  20,   7,   1;
%e A173117   1,  8, 27,  49,  49,  27,   8,   1;
%e A173117   1,  9, 35,  76,  98,  76,  35,   9,  1;
%e A173117   1, 10, 44, 111, 175, 175, 111,  44, 10,  1;
%e A173117   1, 11, 54, 155, 286, 350, 286, 155, 54, 11, 1;
%t A173117 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 A173117 Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 27 2021 *)
%o A173117 (Sage)
%o A173117 @CachedFunction
%o A173117 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 A173117 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 27 2021
%Y A173117 Cf. A007318 (q=0), A072405 (q= -1), this sequence (q=1), A173118 (q=2), A173119 (q=3), A173120 (q= -4), A173122.
%K A173117 nonn,tabl,easy,less
%O A173117 0,5
%A A173117 _Roger L. Bagula_, Feb 10 2010
%E A173117 Edited by _G. C. Greubel_, Apr 27 2021