|
%I #10 Sep 08 2022 08:45:52
%S 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 1,1,43,361,971,971,361,43,1,1,57,652,2511,3876,2511,652,57,1,1,73,
%U 1093,5713,12606,12606,5713,1093,73,1,1,91,1729,11789,35246,50358,35246,11789,1729,91,1
%N Number triangle T(n,k) = Sum_{j=0..n} C(j,n-k)*C(j,k)*(-1)^(n-j).
%C T(n,k) = T(n,n-k).
%C Row sums are A176332.
%C Diagonal sums are A176334.
%C Central coefficients T(2n,n) are A176335.
%H G. C. Greubel, <a href="/A176331/b176331.txt">Rows n = 0..100 of triangle, flattened</a>
%e Triangle begins
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 7, 7, 1;
%e 1, 13, 28, 13, 1;
%e 1, 21, 79, 79, 21, 1;
%e 1, 31, 181, 315, 181, 31, 1;
%e 1, 43, 361, 971, 971, 361, 43, 1;
%e 1, 57, 652, 2511, 3876, 2511, 652, 57, 1;
%e 1, 73, 1093, 5713, 12606, 12606, 5713, 1093, 73, 1;
%e 1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1;
%p T:= proc(n, k) option remember; add( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k), j=0..n); end:
%p seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Dec 07 2019
%t 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 (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 (Magma) T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
%o [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 07 2019
%o (Sage)
%o @CachedFunction
%o def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 07 2019
%o (GAP)
%o T:= function(n,k)
%o return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
%o end;
%o Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Dec 07 2019
%Y Cf. A176332, A176334, A176335.
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Apr 15 2010
|
