|
%I #11 Feb 11 2021 01:54:24
%S 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 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 197,124,33,1,1,36,204,304,197,44,197,304,204,36,1
%N Triangle T(n,k,q) = Sum_{j=0..10} q^j * floor(A174093(n,k)/2^j) with q=2, read by rows.
%C Row sums are: 1, 2, 14, 26, 42, 62, 138, 208, 472, 870, 1528, ...
%H G. C. Greubel, <a href="/A174096/b174096.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 = 2.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 12, 1;
%e 1, 12, 12, 1;
%e 1, 12, 16, 12, 1;
%e 1, 13, 17, 17, 13, 1;
%e 1, 16, 36, 32, 36, 16, 1;
%e 1, 17, 49, 37, 37, 49, 17, 1;
%e 1, 32, 93, 92, 36, 92, 93, 32, 1;
%e 1, 33, 124, 197, 80, 80, 197, 124, 33, 1;
%e 1, 36, 204, 304, 197, 44, 197, 304, 204, 36, 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, 2], {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,2) 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,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 10 2021
%Y Cf. A174093 (q=0), A174095 (q=1), this sequence (q=2), A174097 (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
|
