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%I #7 Feb 26 2021 20:14:04
%S 1,1,1,1,-3,1,1,33,33,1,1,-627,6897,-627,1,1,16929,3538161,3538161,
%T 16929,1,1,-592515,3343562145,-63527680755,3343562145,-592515,1,1,
%U 25478145,5032061028225,2581447307479425,2581447307479425,5032061028225,25478145,1
%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 = 3, read by rows.
%C Row sums are: {1, 2, -1, 68, 5645, 7110182, -56841741493, 5172958787971592, 4953496772756652670937, ...}.
%H G. C. Greubel, <a href="/A156698/b156698.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 = 3.
%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) = (3, 2, -1). - _G. C. Greubel_, Feb 26 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -3, 1;
%e 1, 33, 33, 1;
%e 1, -627, 6897, -627, 1;
%e 1, 16929, 3538161, 3538161, 16929, 1;
%e 1, -592515, 3343562145, -63527680755, 3343562145, -592515, 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,3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 26 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,3,2,-1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 26 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,3,2,-1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 26 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,3,2,-1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 26 2021
%Y Cf. A007318 (m=0), A156696 (m=1), A156697 (m=2), this sequence (m=3).
%Y Cf. A156690, A156691, A156692, A156693.
%Y Cf. A156692, A156699, A156727.
%K sign,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 13 2009
%E Edited by _G. C. Greubel_, Feb 26 2021
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