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A339000 Triangle read by rows: T(n, k) = C(n, k)*Sum_{j=0..n} C(n, k-j)*C(n+j, j)/C(2*j, j).

%I A339000 #25 Jul 31 2024 03:16:50
%S A339000 1,1,2,1,7,5,1,15,32,13,1,26,111,123,34,1,40,285,603,429,89,1,57,610,
%T A339000 2094,2748,1408,233,1,77,1155,5845,12170,11196,4437,610,1,100,2002,
%U A339000 14014,42355,60686,42255,13587,1597,1,126,3246,30030,124137,254756,271961,150951,40736,4181
%N A339000 Triangle read by rows: T(n, k) = C(n, k)*Sum_{j=0..n} C(n, k-j)*C(n+j, j)/C(2*j, j).
%H A339000 G. C. Greubel, <a href="/A339000/b339000.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A339000 G.f.: A008459(x,y)/(1-x*y*A008459(x,y)^2).
%F A339000 T(n,n) = Fibonacci(2*n+1).
%e A339000 Triangle begins as:
%e A339000   1;
%e A339000   1,  2;
%e A339000   1,  7,   5;
%e A339000   1, 15,  32,   13;
%e A339000   1, 26, 111,  123,   34;
%e A339000   1, 40, 285,  603,  429,   89;
%e A339000   1, 57, 610, 2094, 2748, 1408, 233;
%t A339000 T[n_, k_]:= With[{B=Binomial}, B[n,k]*Sum[B[n,k-j]*B[n+j,j]/B[2*j,j], {j,0,n}]];
%t A339000 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 31 2024 *)
%o A339000 (Maxima)
%o A339000 T(n,m):=(binomial(n,m))*sum(((binomial(n,m-k))*(binomial(n+k,k)) )/(binomial(2*k,k)),k,0,n);
%o A339000 (Magma)
%o A339000 b:=Binomial;
%o A339000 A339000:= func< n,k | b(n,k)*(&+[b(n,k-j)*b(n+j,j)/b(2*j,j): j in [0..n]]) >;
%o A339000 [A339000(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 31 2024
%o A339000 (SageMath)
%o A339000 b=binomial
%o A339000 def A339000(n,k): return b(n,k)*sum(b(n,k-j)*b(n+j,j)//b(2*j,j) for j in range(n+1))
%o A339000 flatten([[A339000(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 31 2024
%Y A339000 Cf. A000045 (Fibonacci), A001519, A008459, A046748 (row sums).
%K A339000 nonn,tabl
%O A339000 0,3
%A A339000 _Vladimir Kruchinin_, Nov 18 2020