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A119307 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, k)*C(j, n - k).

%I A119307 #18 May 13 2024 05:11:24
%S A119307 1,1,1,1,5,1,1,11,11,1,1,19,46,19,1,1,29,127,127,29,1,1,41,281,517,
%T A119307 281,41,1,1,55,541,1579,1579,541,55,1,1,71,946,4001,6376,4001,946,71,
%U A119307 1,1,89,1541,8889,20626,20626,8889,1541,89,1,1,109,2377,17907,56904,82994
%N A119307 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, k)*C(j, n - k).
%H A119307 Indranil Ghosh, <a href="/A119307/b119307.txt">Rows 0..100, flattened</a>
%F A119307 T(n, k) = T(n, n - k).
%F A119307 T(n, k) = binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], 1) for k=0..n-1. - _Peter Luschny_, May 13 2024
%e A119307 Triangle begins
%e A119307   1,
%e A119307   1, 1,
%e A119307   1, 5, 1,
%e A119307   1, 11, 11, 1,
%e A119307   1, 19, 46, 19, 1,
%e A119307   1, 29, 127, 127, 29, 1,
%e A119307   1, 41, 281, 517, 281, 41, 1
%e A119307   ...
%p A119307 T := (n, k) -> if n = k then 1 else binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], 1) fi: for n from 0 to 9 do seq(simplify(T(n, k)), k = 0..n) od;
%p A119307 # _Peter Luschny_, May 13 2024
%t A119307 Flatten[Table[Sum[Binomial[j,k] Binomial[j,n-k],{j,0,n}],{n,0,10},{k,0,n}]] (* _Indranil Ghosh_, Mar 03 2017 *)
%o A119307 (PARI)
%o A119307 tabl(nn)={for (n=0, nn, for(k=0, n, print1(sum(j=0, n, binomial(j,k)*binomial(j,n-k)),", ");); print(););};
%o A119307 tabl(10); \\ _Indranil Ghosh_, Mar 03 2017
%Y A119307 Second column is A028387.
%Y A119307 Row sums are A014300.
%Y A119307 Central coefficients T(2*n, n) are A112029.
%K A119307 easy,nonn,tabl
%O A119307 0,5
%A A119307 _Paul Barry_, May 13 2006