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A123736 Triangle T(n,k) = Sum_{j=0..k/2} binomial(n-j-1,k-2*j), read by rows.

%I A123736 #17 Feb 16 2025 08:33:03
%S A123736 1,0,1,1,1,0,1,2,2,1,1,0,1,3,4,3,2,1,1,0,1,4,7,7,5,3,2,1,1,0,1,5,11,
%T A123736 14,12,8,5,3,2,1,1,0,1,6,16,25,26,20,13,8,5,3,2,1,1,0,1,7,22,41,51,46,
%U A123736 33,21,13,8,5,3,2,1,1,0,1,8,29,63,92,97,79,54,34,21,13,8,5,3,2,1,1,0,1
%N A123736 Triangle T(n,k) = Sum_{j=0..k/2} binomial(n-j-1,k-2*j), read by rows.
%C A123736 Row sums give: A000225
%H A123736 G. C. Greubel, <a href="/A123736/b123736.txt">Rows n = 1..100 of triangle, flattened</a>
%H A123736 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SteenrodAlgebra.html">Steenrod Algebra</a>
%e A123736 The triangle starts in row n=1 with columns 0 <= k < 2*n:
%e A123736   1, 0;
%e A123736   1, 1,  1,  0;
%e A123736   1, 2,  2,  1,  1,  0;
%e A123736   1, 3,  4,  3,  2,  1,  1,  0;
%e A123736   1, 4,  7,  7,  5,  3,  2,  1,  1,  0;
%e A123736   1, 5, 11, 14, 12,  8,  5,  3,  2,  1,  1, 0;
%e A123736   1, 6, 16, 25, 26, 20, 13,  8,  5,  3,  2, 1, 1, 0;
%e A123736   1, 7, 22, 41, 51, 46, 33, 21, 13,  8,  5, 3, 2, 1, 1, 0;
%e A123736   1, 8, 29, 63, 92, 97, 79, 54, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0;
%p A123736 seq(seq(sum(binomial(n-j-1, k-2*j), j=0..floor(k/2)), k=0..2*n-1), n=1..10); # _G. C. Greubel_, Sep 05 2019
%t A123736 Table[Sum[Binomial[n-j-1, k-2*j], {j,0,Floor[k/2]}], {n, 10}, {k, 0, 2*n-1}]//Flatten (* modified by _G. C. Greubel_, Sep 05 2019 *)
%o A123736 (PARI) T(n,k) = sum(j=0, k\2, binomial(n-j-1, k-2*j));
%o A123736 for(n=1,10, for(k=0,2*n-1, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Sep 05 2019
%o A123736 (Magma) [&+[Binomial(n-j-1, k-2*j): j in [0..Floor(k/2)]]: k in [0..2*n-1], n in [1..10]]; // _G. C. Greubel_, Sep 05 2019
%o A123736 (Sage) [[sum(binomial(n-j-1, k-2*j) for j in (0..floor(k/2))) for k in (0..2*n-1)] for n in (1..10)] # _G. C. Greubel_, Sep 05 2019
%o A123736 (GAP) Flat(List([1..10], n-> List([0..2*n-1], k-> Sum([0..Int(k/2)], j-> Binomial(n-j-1, k-2*j) )))); # _G. C. Greubel_, Sep 05 2019
%Y A123736 Cf. A136431 (antidiagonals), A027926 (row-reversed), A004006 (column m=3)
%K A123736 nonn,easy,tabf
%O A123736 1,8
%A A123736 _Roger L. Bagula_, Nov 14 2006