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A176331 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, n-k) * C(j, k) * (-1)^(n - j).

%I A176331 #16 May 13 2024 05:12:38
%S A176331 1,1,1,1,3,1,1,7,7,1,1,13,28,13,1,1,21,79,79,21,1,1,31,181,315,181,31,
%T A176331 1,1,43,361,971,971,361,43,1,1,57,652,2511,3876,2511,652,57,1,1,73,
%U A176331 1093,5713,12606,12606,5713,1093,73,1,1,91,1729,11789,35246,50358,35246,11789,1729,91,1
%N A176331 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, n-k) * C(j, k) * (-1)^(n - j).
%H A176331 G. C. Greubel, <a href="/A176331/b176331.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176331 T(n, k) = T(n, n-k).
%F A176331 T(n, k) = binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], -1). - _Peter Luschny_, May 13 2024
%e A176331 Triangle begins
%e A176331   1;
%e A176331   1,  1;
%e A176331   1,  3,    1;
%e A176331   1,  7,    7,     1;
%e A176331   1, 13,   28,    13,     1;
%e A176331   1, 21,   79,    79,    21,     1;
%e A176331   1, 31,  181,   315,   181,    31,     1;
%e A176331   1, 43,  361,   971,   971,   361,    43,     1;
%e A176331   1, 57,  652,  2511,  3876,  2511,   652,    57,    1;
%e A176331   1, 73, 1093,  5713, 12606, 12606,  5713,  1093,   73,  1;
%e A176331   1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1;
%p A176331 T:= proc(n, k) option remember; add( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k), j=0..n); end: seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Dec 07 2019
%p A176331 T := (n, k) -> binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], -1):
%p A176331 seq(seq(simplify(T(n, k)), k = 0..n), n = 0..9); # _Peter Luschny_, May 13 2024
%t A176331 T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 07 2019 *)
%o A176331 (PARI) T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k)); \\ _G. C. Greubel_, Dec 07 2019
%o A176331 (Magma) T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
%o A176331 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 07 2019
%o A176331 (Sage)
%o A176331 @CachedFunction
%o A176331 def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
%o A176331 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 07 2019
%o A176331 (GAP)
%o A176331 T:= function(n,k)
%o A176331     return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
%o A176331   end;
%o A176331 Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Dec 07 2019
%Y A176331 Row sums are A176332.
%Y A176331 Diagonal sums are A176334.
%Y A176331 Central coefficients T(2*n, n) are A176335.
%K A176331 easy,nonn,tabl
%O A176331 0,5
%A A176331 _Paul Barry_, Apr 15 2010