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A120247 Triangle of Hankel transforms of binomial(n+k, k).

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