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A174095 Triangle T(n,k,q) = Sum_{j=0..10} q^j * floor(A174093(n,k)/2^j) with q=1, read by rows.

%I A174095 #13 Feb 11 2021 01:55:13
%S A174095 1,1,1,1,7,1,1,7,7,1,1,7,10,7,1,1,8,11,11,8,1,1,10,18,15,18,10,1,1,11,
%T A174095 26,19,19,26,11,1,1,15,39,38,18,38,39,15,1,1,16,53,67,31,31,67,53,16,
%U A174095 1,1,18,70,109,67,22,67,109,70,18,1
%N A174095 Triangle T(n,k,q) = Sum_{j=0..10} q^j * floor(A174093(n,k)/2^j) with q=1, read by rows.
%C A174095 Row sums are: 1, 2, 9, 16, 26, 40, 73, 114, 204, 336, 552, ...
%H A174095 G. C. Greubel, <a href="/A174095/b174095.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A174095 T(n, k, q) = Sum_{j=0..10} q^j * floor(A174093(n, k)/2^j), for q = 1.
%e A174095 Triangle begins as:
%e A174095   1;
%e A174095   1,  1;
%e A174095   1,  7,  1;
%e A174095   1,  7,  7,   1;
%e A174095   1,  7, 10,   7,  1;
%e A174095   1,  8, 11,  11,  8,  1;
%e A174095   1, 10, 18,  15, 18, 10,  1;
%e A174095   1, 11, 26,  19, 19, 26, 11,   1;
%e A174095   1, 15, 39,  38, 18, 38, 39,  15,  1;
%e A174095   1, 16, 53,  67, 31, 31, 67,  53, 16,  1;
%e A174095   1, 18, 70, 109, 67, 22, 67, 109, 70, 18, 1;
%t A174095 A174093[n_, k_]:= If[n<2, 1, Binomial[n-k+1, k] + Binomial[k+1, n-k]];
%t A174095 T[n_, k_, q_]:= Sum[q^j*Floor[A174093[n, k]/2^j], {j, 0, 10}];
%t A174095 Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Feb 10 2021 *)
%o A174095 (Sage)
%o A174095 def A174093(n,k): return 1 if n<2 else binomial(n-k+1, k) + binomial(k+1, n-k)
%o A174095 def T(n,k,q): return sum( q^j*(A174093(n,k)//2^j) for j in (0..10) )
%o A174095 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 10 2021
%o A174095 (Magma)
%o A174095 A174093:= func< n,k | n lt 2 select 1 else Binomial(n-k+1, k) + Binomial(k+1, n-k) >;
%o A174095 T:= func< n,k,q | (&+[ q^j*Floor(A174093(n,k)/2^j): j in [0..10]]) >;
%o A174095 [T(n,k,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 10 2021
%Y A174095 Cf. A174093 (q=0), this sequence (q=1), A174096 (q=2), A174097 (q=3).
%K A174095 nonn,tabl,easy,less
%O A174095 0,5
%A A174095 _Roger L. Bagula_, Mar 07 2010
%E A174095 Edited by _G. C. Greubel_, Feb 10 2021