A119307 - OEIS

%I #18 May 13 2024 05:11:24

%S 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 281,41,1,1,55,541,1579,1579,541,55,1,1,71,946,4001,6376,4001,946,71,

%U 1,1,89,1541,8889,20626,20626,8889,1541,89,1,1,109,2377,17907,56904,82994

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

%H Indranil Ghosh, <a href="/A119307/b119307.txt">Rows 0..100, flattened</a>

%F T(n, k) = T(n, n - k).

%F 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 Triangle begins

%e   1,

%e   1, 1,

%e   1, 5, 1,

%e   1, 11, 11, 1,

%e   1, 19, 46, 19, 1,

%e   1, 29, 127, 127, 29, 1,

%e   1, 41, 281, 517, 281, 41, 1

%e   ...

%p 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 # _Peter Luschny_, May 13 2024

%t 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 (PARI)

%o tabl(nn)={for (n=0, nn, for(k=0, n, print1(sum(j=0, n, binomial(j,k)*binomial(j,n-k)),", ");); print(););};

%o tabl(10); \\ _Indranil Ghosh_, Mar 03 2017

%Y Second column is A028387.

%Y Row sums are A014300.

%Y Central coefficients T(2*n, n) are A112029.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, May 13 2006