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A173076 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 = 2, read by rows.

%I A173076 #6 Sep 08 2022 08:45:50
%S A173076 1,1,1,1,3,1,1,4,4,1,1,7,13,7,1,1,8,21,21,8,1,1,13,46,67,46,13,1,1,14,
%T A173076 60,114,114,60,14,1,1,23,123,295,389,295,123,23,1,1,24,147,419,685,
%U A173076 685,419,147,24,1,1,41,300,1015,2001,2491,2001,1015,300,41,1
%N A173076 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 = 2, read by rows.
%H A173076 G. C. Greubel, <a href="/A173076/b173076.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A173076 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 = 2.
%e A173076 Triangle begins as:
%e A173076   1;
%e A173076   1,  1;
%e A173076   1,  3,   1;
%e A173076   1,  4,   4,    1;
%e A173076   1,  7,  13,    7,    1;
%e A173076   1,  8,  21,   21,    8,    1;
%e A173076   1, 13,  46,   67,   46,   13,    1;
%e A173076   1, 14,  60,  114,  114,   60,   14,    1;
%e A173076   1, 23, 123,  295,  389,  295,  123,   23,   1;
%e A173076   1, 24, 147,  419,  685,  685,  419,  147,  24,  1;
%e A173076   1, 41, 300, 1015, 2001, 2491, 2001, 1015, 300, 41, 1;
%t A173076 T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(Floor[n/2])*Binomial[n-2, k-1]];
%t A173076 Table[T[n, k, 2], {n,0,10}, {k,0,n}]//Flatten
%o A173076 (Magma)
%o A173076 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 A173076 [T(n,k,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 09 2021
%o A173076 (Sage)
%o A173076 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 A173076 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)])
%Y A173076 Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173077 (q=3).
%K A173076 nonn,tabl
%O A173076 0,5
%A A173076 _Roger L. Bagula_, Feb 09 2010
%E A173076 Edited by _G. C. Greubel_, Jul 09 2021