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%I #19 Aug 04 2022 05:11:50
%S 1,1,1,1,1,2,1,1,3,3,1,1,4,4,5,1,1,5,5,8,8,1,1,6,6,12,12,13,1,1,7,7,
%T 17,17,21,21,1,1,8,8,23,23,33,33,34,1,1,9,9,30,30,50,50,55,55,1,1,10,
%U 10,38,38,73,73,88,88,89,1,1,11,11,47,47,103,103,138,138,144,144,1,1,12,12,57,57,141,141,211,211,232,232,233
%N Reverse of number triangle A054450.
%H G. C. Greubel, <a href="/A099573/b099573.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Number triangle T(n, k) = Sum_{j=0..floor(k/2)} binomial(n-j, j) if k <= n, 0 otherwise.
%F T(n, n) = A000045(n+1).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A099574(n).
%F Sum_{k=0..n} T(n, k) = A029907(n+1).
%F Antidiagonals of the following array: the first row equals the Fibonacci numbers, (1, 1, 2, 3, 5, ...), and the (n+1)-st row is obtained by the matrix-vector product A128174 * n-th row. - _Gary W. Adamson_, Jan 19 2011
%F From _G. C. Greubel_, Jul 25 2022: (Start)
%F T(n, n-1) = A052952(n-1), n >= 1.
%F T(n, n-2) = A054451(n-2), n >= 2.
%F T(n, n-3) = A099571(n-3), n >= 3.
%F T(n, n-4) = A099572(n-4), n >= 4. (End)
%e First few rows of the array:
%e 1, 1, 2, 3, 5, 8, ... (A000045)
%e 1, 1, 3, 4, 8, 12, ... (A052952)
%e 1, 1, 4, 5, 12, 17, ... (A054451)
%e 1, 1, 5, 6, 17, 23, ... (A099571)
%e 1, 1, 6, 7, 23, 30, ... (A099572)
%e ...
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 1, 2;
%e 1, 1, 3, 3;
%e 1, 1, 4, 4, 5;
%e 1, 1, 5, 5, 8, 8;
%e 1, 1, 6, 6, 12, 12, 13;
%e 1, 1, 7, 7, 17, 17, 21, 21;
%e 1, 1, 8, 8, 23, 23, 33, 33, 34;
%e 1, 1, 9, 9, 30, 30, 50, 50, 55, 55;
%t T[n_, k_]:= Sum[Binomial[n-j,j], {j,0,Floor[k/2]}];
%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 25 2022 *)
%o (Magma) [(&+[Binomial(n-j,j): j in [0..Floor(k/2)]]): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jul 25 2022
%o (SageMath)
%o def A099573(n,k): return sum(binomial(n-j, j) for j in (0..(k//2)))
%o flatten([[A099573(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jul 25 2022
%Y Cf. A000045, A029907 (row sums), A052951, A052952, A054450, A054451, A052952.
%Y Cf. A099571, A099572, A099574 (diagonal sums), A099575.
%K easy,nonn,tabl
%O 0,6
%A _Paul Barry_, Oct 23 2004
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