%I #11 Mar 01 2021 02:01:44

%S 1,1,1,1,-2,1,1,16,16,1,1,-224,1792,-224,1,1,4480,501760,501760,4480,

%T 1,1,-116480,260915200,-3652812800,260915200,-116480,1,1,3727360,

%U 217081446400,60782804992000,60782804992000,217081446400,3727360,1

%N 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 Row sums are: {1, 2, 0, 34, 1346, 1012482, -3131215358, 121999780331522,

%C 34591292869081661442, 107137531255480378706493442, ...}.

%H G. C. Greubel, <a href="/A156697/b156697.txt">Rows n = 0..30 of the triangle, flattened</a>

%F 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 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 Triangle begins as:

%e 1;

%e 1, 1;

%e 1, -2, 1;

%e 1, 16, 16, 1;

%e 1, -224, 1792, -224, 1;

%e 1, 4480, 501760, 501760, 4480, 1;

%e 1, -116480, 260915200, -3652812800, 260915200, -116480, 1;

%t (* First program *)

%t t[n_, k_]:= If[k==0, n!, Product[1 -(2*i-1)*(k+1), {j,n}, {i,0,j-1}] ];

%t T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];

%t Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 25 2021 *)

%t (* Second program *)

%t f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j,n}];

%t 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 Table[T[n,k,2,2,-1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 25 2021 *)

%o (Sage)

%o @CachedFunction

%o def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))

%o 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 flatten([[T(n,k,2,2,-1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 25 2021

%o (Magma)

%o 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 T:= func< n,k,m,p,q | f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q)) >;

%o [T(n,k,2,2,-1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 25 2021

%Y Cf. A007318 (m=0), A156696 (m=1), this sequence (m=2), A156698 (m=3).

%Y Cf. A156690, A156691, A156692, A156693.

%Y Cf. A156691, A156699, A156725.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 13 2009

%E Edited by _G. C. Greubel_, Feb 25 2021