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%I #14 Sep 08 2022 08:45:50
%S 1,1,1,1,4,1,1,5,5,1,1,12,23,12,1,1,13,36,36,13,1,1,32,122,181,122,32,
%T 1,1,33,155,304,304,155,33,1,1,88,513,1270,1689,1270,513,88,1,1,89,
%U 602,1784,2960,2960,1784,602,89,1,1,252,1988,6923,13817,17261,13817,6923,1988,252,1
%N Triangle T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.
%H G. C. Greubel, <a href="/A173077/b173077.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3.
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 5, 5, 1;
%e 1, 12, 23, 12, 1;
%e 1, 13, 36, 36, 13, 1;
%e 1, 32, 122, 181, 122, 32, 1;
%e 1, 33, 155, 304, 304, 155, 33, 1;
%e 1, 88, 513, 1270, 1689, 1270, 513, 88, 1;
%e 1, 89, 602, 1784, 2960, 2960, 1784, 602, 89, 1;
%e 1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1;
%e ...
%e Row sums: 1, 2, 6, 12, 49, 100, 491, 986, 5433, 10872, 63223, ...
%t T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + 3^Floor[n/2] Binomial[n-2, k- 1]];
%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o (Magma)
%o T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) + q^(Floor(n/2))*Binomial(n-2,k-1) -1 >;
%o [T(n,k,3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 09 2021
%o (Sage)
%o def T(n,k,q): return 1 if (k==0 or k==n) else binomial(n,k) + q^(n//2)*binomial(n-2,k-1) -1
%o flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 09 2021
%Y Cf. A132044 (q=0), A173075 (q=1), A173076 (q=2), this sequence (q=3).
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 09 2010
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