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A174093 Triangle T(n, k) = binomial(n-k+1, k) + binomial(k+1, n-k) with T(0,0) = T(1, 0) = T(1, 1) = 1, read by rows.

%I A174093 #12 Feb 11 2021 02:45:29
%S A174093 1,1,1,1,4,1,1,4,4,1,1,4,6,4,1,1,5,7,7,5,1,1,6,10,8,10,6,1,1,7,15,11,
%T A174093 11,15,7,1,1,8,21,20,10,20,21,8,1,1,9,28,35,16,16,35,28,9,1,1,10,36,
%U A174093 56,35,12,35,56,36,10,1
%N A174093 Triangle T(n, k) = binomial(n-k+1, k) + binomial(k+1, n-k) with T(0,0) = T(1, 0) = T(1, 1) = 1, read by rows.
%H A174093 G. C. Greubel, <a href="/A174093/b174093.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A174093 T(n, k) = binomial(n-k+1, k) + binomial(k+1, n-k) with T(0,0) = T(1, 0) = T(1, 1) = 1.
%F A174093 From _G. C. Greubel_, Feb 10 2021: (Start)
%F A174093 T(n, k) = A007318(n-k+1, k) + A007318(k+1, n-k), for rows n >= 2.
%F A174093 T(n, k) = T(n, n-k).
%F A174093 Sum_{k=0..n} T(n, k) = 2*Fibonacci(n+2) - [n=0] - 2*[n=1] = 2*A071679(n) + [n=0], where [] is the Iverson bracket. (End)
%e A174093 Triangle begins as:
%e A174093   1;
%e A174093   1,  1;
%e A174093   1,  4,  1;
%e A174093   1,  4,  4,  1;
%e A174093   1,  4,  6,  4,  1;
%e A174093   1,  5,  7,  7,  5,  1;
%e A174093   1,  6, 10,  8, 10,  6,  1;
%e A174093   1,  7, 15, 11, 11, 15,  7,  1;
%e A174093   1,  8, 21, 20, 10, 20, 21,  8,  1;
%e A174093   1,  9, 28, 35, 16, 16, 35, 28,  9,  1;
%e A174093   1, 10, 36, 56, 35, 12, 35, 56, 36, 10, 1;
%t A174093 T[n_, k_]:= If[n==0 || n==1, 1, Binomial[n-k+1, k] + Binomial[k+1, n-k]];
%t A174093 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
%o A174093 (Sage)
%o A174093 def T(n, k):
%o A174093     if (n==0 or n==1): return 1
%o A174093     else: return binomial(n-k+1, k) + binomial(k+1, n-k)
%o A174093 flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 10 2021
%o A174093 (Magma)
%o A174093 T:= func< n,k | n lt 2 select 1 else Binomial(n-k+1, k) + Binomial(k+1, n-k) >;
%o A174093 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 10 2021
%Y A174093 Cf. A000045, A007318, A071679.
%Y A174093 Cf. A174095, A174096, A174097.
%K A174093 nonn,tabl,easy,less
%O A174093 0,5
%A A174093 _Roger L. Bagula_, Mar 07 2010
%E A174093 Edited by _G. C. Greubel_, Feb 10 2021