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%I #34 Aug 28 2024 14:58:57
%S 1,0,2,0,0,5,0,0,1,14,0,0,1,8,42,0,0,1,10,45,132,0,0,1,12,69,220,429,
%T 0,0,1,14,98,406,1001,1430,0,0,1,16,132,672,2184,4368,4862,0,0,1,18,
%U 171,1032,4152,11088,18564,16796,0,0,1,20,215,1500,7185,23904,54060,77520,58786
%N Binomial transform of a signed variant of triangle A050166.
%C Row sums = A007317(n+1).
%C Right border = A000108(n+1).
%H Alois P. Heinz, <a href="/A329959/b329959.txt">Rows n = 0..140, flattened</a>
%F T(n,k) = Sum_{j=k..n} C(n,j) * (-1)^(j+k) * A050166(j,k). - _Alois P. Heinz_, Nov 27 2019
%e The signed variant of A050166 is A050166(n,k) * (-1)^(n+k):
%e 1;
%e -1, 2;
%e 1, -4, 5;
%e -1, 6, -14, 14;
%e 1, -8, 27, -48, 42;
%e ...
%e Let the above triangle = S, and Pascal's triangle = P as an infinite lower triangular matrix. Then T = P * S gives:
%e 1;
%e 0, 2;
%e 0, 0, 5;
%e 0, 0, 1, 14;
%e 0, 0, 1, 8, 42;
%e 0, 0, 1, 10, 45, 132;
%e ...
%p S:= (n, k)-> (binomial(2*n, k)-binomial(2*n, k-2))*(-1)^(n+k):
%p T:= (n, k)-> add(binomial(n, j)*S(j, k), j=k..n):
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Nov 27 2019
%t Table[Sum[(-1)^(k+j)*Binomial[n, j]*(Binomial[2*j, k] - Binomial[2*j, k-2]), {j, k, n}], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 06 2020 *)
%o (PARI) T(n,k) = sum(j=k, n, (-1)^(k+j)*binomial(n,j)*(binomial(2*j,k) - binomial(2*j,k-2)) ); \\ _G. C. Greubel_, Jan 06 2020
%o (Magma) T:= func< n,k | &+[(-1)^(k+j)*Binomial(n,j)*(Binomial(2*j,k) - Binomial(2*j,k-2)): j in [k..n]] >;
%o [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jan 06 2020
%o (Sage) [[sum((-1)^(k+j)*binomial(n,j)*(binomial(2*j,k) - binomial(2*j,k-2)) for j in (k..n)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jan 06 2020
%o (GAP) B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> Sum([k..n], j-> (-1)^(k+j)*B(n,j)*(B(2*j,k) - B(2*j,k-2)) )))); # _G. C. Greubel_, Jan 06 2020
%Y Cf. A000108, A007317, A007318, A050166.
%K nonn,tabl,changed
%O 0,3
%A _Gary W. Adamson_, Nov 25 2019
%E New offset and more terms from _Alois P. Heinz_, Nov 25 2019
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