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%I A156697 #11 Mar 01 2021 02:01:44
%S A156697 1,1,1,1,-2,1,1,16,16,1,1,-224,1792,-224,1,1,4480,501760,501760,4480,
%T A156697 1,1,-116480,260915200,-3652812800,260915200,-116480,1,1,3727360,
%U A156697 217081446400,60782804992000,60782804992000,217081446400,3727360,1
%N A156697 Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2*i-1) ) and m = 2, read by rows.
%C A156697 Row sums are: {1, 2, 0, 34, 1346, 1012482, -3131215358, 121999780331522,
%C A156697 34591292869081661442, 107137531255480378706493442, ...}.
%H A156697 G. C. Greubel, <a href="/A156697/b156697.txt">Rows n = 0..30 of the triangle, flattened</a>
%F A156697 T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2*i-1) ) and m = 2.
%F A156697 T(n, k, m, p, q) = (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q))) where Product_{j=1..n} Pochhammer( (q*(m+1) -1)/(p*(m+1)), j) for (m, p, q) = (2, 2, -1). - _G. C. Greubel_, Feb 25 2021
%e A156697 Triangle begins as:
%e A156697   1;
%e A156697   1,       1;
%e A156697   1,      -2,         1;
%e A156697   1,      16,        16,           1;
%e A156697   1,    -224,      1792,        -224,         1;
%e A156697   1,    4480,    501760,      501760,      4480,       1;
%e A156697   1, -116480, 260915200, -3652812800, 260915200, -116480, 1;
%t A156697 (* First program *)
%t A156697 t[n_, k_]:= If[k==0, n!, Product[1 -(2*i-1)*(k+1), {j,n}, {i,0,j-1}] ];
%t A156697 T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];
%t A156697 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 25 2021 *)
%t A156697 (* Second program *)
%t A156697 f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j,n}];
%t A156697 T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n,m,p,q]/(f[k,m,p,q]*f[n-k,m,p,q]));
%t A156697 Table[T[n,k,2,2,-1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 25 2021 *)
%o A156697 (Sage)
%o A156697 @CachedFunction
%o A156697 def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))
%o A156697 def T(n, k, m, p, q): return (-p*(m+1))^(k*(n-k))*(f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q)))
%o A156697 flatten([[T(n,k,2,2,-1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 25 2021
%o A156697 (Magma)
%o A156697 f:= func< n,m,p,q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;
%o A156697 T:= func< n,k,m,p,q | f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q)) >;
%o A156697 [T(n,k,2,2,-1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 25 2021
%Y A156697 Cf. A007318 (m=0), A156696 (m=1), this sequence (m=2), A156698 (m=3).
%Y A156697 Cf. A156690, A156691, A156692, A156693.
%Y A156697 Cf. A156691, A156699, A156725.
%K A156697 sign,tabl
%O A156697 0,5
%A A156697 _Roger L. Bagula_, Feb 13 2009
%E A156697 Edited by _G. C. Greubel_, Feb 25 2021

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