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%I #12 Feb 11 2021 02:45:24
%S 1,1,1,1,19,1,1,19,19,1,1,19,24,19,1,1,20,25,25,20,1,1,24,70,65,70,24,
%T 1,1,25,90,71,71,90,25,1,1,65,231,230,70,230,231,65,1,1,66,295,671,
%U 211,211,671,295,66,1,1,70,684,941,671,84,671,941,684,70,1
%N Triangle T(n,k,q) = Sum_{j=0..10} q^j * floor(A174093(n,k)/2^j) with q=3, read by rows.
%C Row sums are: 1, 2, 21, 40, 64, 92, 255, 374, 1124, 2488, 4818, ...
%H G. C. Greubel, <a href="/A174097/b174097.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(n, k, q) = Sum_{j=0..10} q^j * floor(A174093(n, k)/2^j), for q = 3.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 19, 1;
%e 1, 19, 19, 1;
%e 1, 19, 24, 19, 1;
%e 1, 20, 25, 25, 20, 1;
%e 1, 24, 70, 65, 70, 24, 1;
%e 1, 25, 90, 71, 71, 90, 25, 1;
%e 1, 65, 231, 230, 70, 230, 231, 65, 1;
%e 1, 66, 295, 671, 211, 211, 671, 295, 66, 1;
%e 1, 70, 684, 941, 671, 84, 671, 941, 684, 70, 1;
%t A174093[n_, k_]:= If[n<2, 1, Binomial[n-k+1, k] + Binomial[k+1, n-k]];
%t T[n_, k_, q_]:= Sum[q^j*Floor[A174093[n, k]/2^j], {j, 0, 10}];
%t Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Feb 10 2021 *)
%o (Sage)
%o def A174093(n,k): return 1 if n<2 else binomial(n-k+1, k) + binomial(k+1, n-k)
%o def T(n,k,q): return sum( q^j*(A174093(n,k)//2^j) for j in (0..10) )
%o flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 10 2021
%o (Magma)
%o A174093:= func< n,k | n lt 2 select 1 else Binomial(n-k+1, k) + Binomial(k+1, n-k) >;
%o T:= func< n,k,q | (&+[ q^j*Floor(A174093(n,k)/2^j): j in [0..10]]) >;
%o [T(n,k,3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 10 2021
%Y Cf. A174093 (q=0), A174095 (q=1), A174096 (q=2), this sequence (q=3).
%K nonn,tabl,easy,less
%O 0,5
%A _Roger L. Bagula_, Mar 07 2010
%E Edited by _G. C. Greubel_, Feb 10 2021
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