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A174096 Triangle T(n,k,q) = Sum_{j=0..10} q^j * floor(A174093(n,k)/2^j) with q=2, read by rows.

%I A174096 #11 Feb 11 2021 01:54:24
%S A174096 1,1,1,1,12,1,1,12,12,1,1,12,16,12,1,1,13,17,17,13,1,1,16,36,32,36,16,
%T A174096 1,1,17,49,37,37,49,17,1,1,32,93,92,36,92,93,32,1,1,33,124,197,80,80,
%U A174096 197,124,33,1,1,36,204,304,197,44,197,304,204,36,1
%N A174096 Triangle T(n,k,q) = Sum_{j=0..10} q^j * floor(A174093(n,k)/2^j) with q=2, read by rows.
%C A174096 Row sums are: 1, 2, 14, 26, 42, 62, 138, 208, 472, 870, 1528, ...
%H A174096 G. C. Greubel, <a href="/A174096/b174096.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A174096 T(n, k, q) = Sum_{j=0..10} q^j * floor(A174093(n, k)/2^j), for q = 2.
%e A174096 Triangle begins as:
%e A174096   1;
%e A174096   1,  1;
%e A174096   1, 12,   1;
%e A174096   1, 12,  12,   1;
%e A174096   1, 12,  16,  12,   1;
%e A174096   1, 13,  17,  17,  13,  1;
%e A174096   1, 16,  36,  32,  36, 16,   1;
%e A174096   1, 17,  49,  37,  37, 49,  17,   1;
%e A174096   1, 32,  93,  92,  36, 92,  93,  32,   1;
%e A174096   1, 33, 124, 197,  80, 80, 197, 124,  33,  1;
%e A174096   1, 36, 204, 304, 197, 44, 197, 304, 204, 36, 1;
%t A174096 A174093[n_, k_]:= If[n<2, 1, Binomial[n-k+1, k] + Binomial[k+1, n-k]];
%t A174096 T[n_, k_, q_]:= Sum[q^j*Floor[A174093[n, k]/2^j], {j, 0, 10}];
%t A174096 Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Feb 10 2021 *)
%o A174096 (Sage)
%o A174096 def A174093(n,k): return 1 if n<2 else binomial(n-k+1, k) + binomial(k+1, n-k)
%o A174096 def T(n,k,q): return sum( q^j*(A174093(n,k)//2^j) for j in (0..10) )
%o A174096 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 10 2021
%o A174096 (Magma)
%o A174096 A174093:= func< n,k | n lt 2 select 1 else Binomial(n-k+1, k) + Binomial(k+1, n-k) >;
%o A174096 T:= func< n,k,q | (&+[ q^j*Floor(A174093(n,k)/2^j): j in [0..10]]) >;
%o A174096 [T(n,k,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 10 2021
%Y A174096 Cf. A174093 (q=0), A174095 (q=1), this sequence (q=2), A174097 (q=3).
%K A174096 nonn,tabl,easy,less
%O A174096 0,5
%A A174096 _Roger L. Bagula_, Mar 07 2010
%E A174096 Edited by _G. C. Greubel_, Feb 10 2021