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A253273 Triangle T(n,k) = Sum_{j=0..n-k+1} binomial(k+j,k-j+1)*binomial(n-k,j-1), read by rows.

%I A253273 #19 Apr 17 2021 21:48:27
%S A253273 1,1,2,1,3,3,1,4,7,4,1,5,12,14,5,1,6,18,30,25,6,1,7,25,53,66,41,7,1,8,
%T A253273 33,84,136,132,63,8,1,9,42,124,244,315,245,92,9,1,10,52,174,400,636,
%U A253273 673,428,129,10,1,11,63,235,615,1152,1522,1346,711,175,11
%N A253273 Triangle T(n,k) = Sum_{j=0..n-k+1} binomial(k+j,k-j+1)*binomial(n-k,j-1), read by rows.
%H A253273 G. C. Greubel, <a href="/A253273/b253273.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A253273 T(n,k) = Sum_{j=0..n-k+1} binomial(k+j,k-j+1)*binomial(n-k,j-1).
%F A253273 Sum_{k=0..n} T(n,k) = A095263(n+1).
%F A253273 G.f.: 1/( (1-x)*(1+y^2) - (2-x)*y ).
%e A253273 The triangle begins as:
%e A253273   1;
%e A253273   1,  2;
%e A253273   1,  3,  3;
%e A253273   1,  4,  7,   4;
%e A253273   1,  5, 12,  14,   5;
%e A253273   1,  6, 18,  30,  25,   6;
%e A253273   1,  7, 25,  53,  66,  41,   7;
%e A253273   1,  8, 33,  84, 136, 132,  63,   8;
%e A253273   1,  9, 42, 124, 244, 315, 245,  92,   9;
%e A253273   1, 10, 52, 174, 400, 636, 673, 428, 129, 10;
%e A253273   ...
%t A253273 T[n_, k_]:= Sum[Binomial[k+j,k-j+1]*Binomial[n-k,j-1], {j,0,n-k+1}];
%t A253273 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 17 2021 *)
%o A253273 (Maxima)
%o A253273 T(n,m):=sum(binomial(m+k,m-k+1)*binomial(n-m,k-1),k,0,n-m+1);
%o A253273 (Magma)
%o A253273 T:= func< n,k | (&+[Binomial(k+j,k-j+1)*Binomial(n-k,j-1): j in [0..n-k+1]]) >;
%o A253273 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 17 2021
%o A253273 (Sage)
%o A253273 def T(n,k): return sum(binomial(k+j,k-j+1)*binomial(n-k,j-1) for j in (0..n-k+1))
%o A253273 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 17 2021
%Y A253273 Cf. A095263.
%K A253273 nonn,tabl
%O A253273 0,3
%A A253273 _Vladimir Kruchinin_, May 01 2015