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A026009 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(1,1) = 1; and for n >= 2, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[(n+1)/2]; T(n,n/2 + 1) = T(n-1,n/2) if n is even.

%I A026009 #22 Mar 18 2021 23:31:50
%S A026009 1,1,1,1,2,1,1,3,3,1,4,6,3,1,5,10,9,1,6,15,19,9,1,7,21,34,28,1,8,28,
%T A026009 55,62,28,1,9,36,83,117,90,1,10,45,119,200,207,90,1,11,55,164,319,407,
%U A026009 297,1,12,66,219,483,726,704,297,1,13,78,285,702,1209,1430,1001,1,14,91,363,987,1911,2639,2431,1001
%N A026009 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(1,1) = 1; and for n >= 2, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[(n+1)/2]; T(n,n/2 + 1) = T(n-1,n/2) if n is even.
%H A026009 G. C. Greubel, <a href="/A026009/b026009.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A026009 T(n, k) = binomial(n, k) - binomial(n, k-3). - Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001
%F A026009 Sum_{k=0..floor((n+2)/2)} T(n, k) = A026010(n). - _G. C. Greubel_, Mar 18 2021
%e A026009 From _Jonathon Kirkpatrick_, Jul 01 2016: (Start)
%e A026009 Triangle begins:
%e A026009   1;
%e A026009   1,  1;
%e A026009   1,  2,  1;
%e A026009   1,  3,  3;
%e A026009   1,  4,  6,   3;
%e A026009   1,  5, 10,   9;
%e A026009   1,  6, 15,  19,   9;
%e A026009   1,  7, 21,  34,  28;
%e A026009   1,  8, 28,  55,  62,   28;
%e A026009   1,  9, 36,  83, 117,   90;
%e A026009   1, 10, 45, 119, 200,  207,   90;
%e A026009   1, 11, 55, 164, 319,  407,  297;
%e A026009   1, 12, 66, 219, 483,  726,  704,  297;
%e A026009   1, 13, 78, 285, 702, 1209, 1430, 1001;
%e A026009   ... (End)
%t A026009 T[n_, k_]:= Binomial[n, k] - Binomial[n, k-3];
%t A026009 Join[{1}, Table[T[n, k], {n,14}, {k,0,Floor[(n+2)/2]}]//Flatten] (* _G. C. Greubel_, Mar 18 2021 *)
%o A026009 (Sage) [1]+flatten([[binomial(n,k) - binomial(n,k-3) for k in (0..(n+2)//2)] for n in (1..15)]) # _G. C. Greubel_, Mar 18 2021
%o A026009 (Magma) [1] cat [Binomial(n,k) - Binomial(n,k-3): k in [0..Floor((n+2)/2)], n in [1..15]]; // _G. C. Greubel_, Mar 18 2021
%Y A026009 Diagonals of this sequence: A000217, A000245, A026012, A026013, A026014, A026015, A026016, A026017, A026018, A026019, A026020, A026021.
%Y A026009 Sums involving this sequence: A026010, A027287, A027288, A027289, A027290, A027291, A027292.
%K A026009 nonn,tabf,easy
%O A026009 0,5
%A A026009 _Clark Kimberling_