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%I #8 Mar 15 2023 07:54:13
%S 1,1,-1,1,-3,-1,1,-6,-10,1,1,-10,-50,35,1,1,-15,-175,490,126,-1,1,-21,
%T -490,4116,5292,-462,-1,1,-28,-1176,24696,116424,-60984,-1716,1,1,-36,
%U -2520,116424,1646568,-3737448,-736164,6435,1,1,-45,-4950,457380,16818516,-133613766,-131589315,9202050,24310,-1
%N Triangle of Hankel transforms of binomial(n+k, k).
%C Columns include -A000217, -A006542, A107915.
%C Row k is the Hankel transform of C(n+k, k).
%C The matrix inverse starts
%C 1;
%C 1, -1;
%C -2, 3, -1;
%C -15, 24, -10, 1;
%C 434, -700, 300, -35, 1;
%C 47670, -76950, 33075, -3920, 126, -1;
%C -19787592, 31943835, -13733720, 1629936, -52920, 462, -1; - _R. J. Mathar_, Mar 22 2013
%H G. C. Greubel, <a href="/A120247/b120247.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = (cos(pi*k/2) - sin(pi*k/2))*( Product{j=0..k-1} C(n+j+1, k+1)/Product{j=0..k-1} C(k+j+1, k+1) ).
%e Triangle begins
%e 1;
%e 1, -1;
%e 1, -3, -1;
%e 1, -6, -10, 1;
%e 1, -10, -50, 35, 1;
%e 1, -15, -175, 490, 126, -1;
%e 1, -21, -490, 4116, 5292, -462, -1;
%e 1, -28, -1176, 24696, 116424, -60984, -1716, 1;
%p A120247 := proc(n,k)
%p (cos(Pi*k/2)-sin(Pi*k/2))*mul(binomial(n+j+1,k+1),j=0..k-1)/mul(binomial(k+j+1,k+1),j=0..k-1) ;
%p simplify(%) ;
%p end proc: # _R. J. Mathar_, Mar 22 2013
%t p[m_, k_]:= Product[Binomial[m+j, k+1], {j,k}];
%t T[n_, k_]:= (-1)^Floor[(k+1)/2]*p[n,k]/p[k,k];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 15 2023 *)
%o (Magma)
%o p:= func< m,k | k eq 0 select 1 else (&*[Binomial(m+j, k+1): j in [1..k]]) >;
%o A120247:= func< n,k | (-1)^Floor((k+1)/2)*p(n,k)/p(k,k) >;
%o [A120247(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 15 2023
%o (SageMath)
%o def p(m,k): return product(binomial(m+j+1,k+1) for j in range(k))
%o def A120247(n,k): return (-1)^((k+1)//2)*p(n,k)/p(k,k)
%o flatten([[A120247(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Mar 15 2023
%Y Columns include: A000217, A006542, A107915.
%Y Cf. A120248.
%K easy,sign,tabl
%O 0,5
%A _Paul Barry_, Jun 12 2006
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