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%I #12 Jun 25 2021 03:35:33
%S 1,1,1,1,-1,1,1,1,1,1,1,1,-1,1,1,1,-2,2,2,-2,1,1,1,2,2,2,1,1,1,1,-1,2,
%T 2,-1,1,1,1,-2,2,2,-4,2,2,-2,1,1,1,2,2,2,2,2,2,1,1,1,1,-1,2,2,-1,2,2,
%U -1,1,1
%N Triangle T(n, k, m) = round( t(n,m)/(t(k,m)*t(n-k,m)) ), with T(0, k, m) = 1, where t(n, k) = Product_{j=1..n} A129862(k+1, j), t(n, 0) = n!, and m = 2, read by rows.
%C The original definition of this sequence said it was based on the Cartan matrix of type D_n, so that matrix is somehow implicitly involved. - _N. J. A. Sloane_, Jun 25 2021
%H G. C. Greubel, <a href="/A156608/b156608.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(n, k, m) = round( t(n,m)/(t(k,m)*t(n-k,m)) ), with T(0, k, m) = 1, where t(n, k) = Product_{j=1..n} A129862(k+1, j), t(n, 0) = n!, and m = 2.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, -1, 1;
%e 1, 1, 1, 1;
%e 1, 1, -1, 1, 1;
%e 1, -2, 2, 2, -2, 1;
%e 1, 1, 2, 2, 2, 1, 1;
%e 1, 1, -1, 2, 2, -1, 1, 1;
%e 1, -2, 2, 2, -4, 2, 2, -2, 1;
%e 1, 1, 2, 2, 2, 2, 2, 2, 1, 1;
%e 1, 1, -1, 2, 2, -1, 2, 2, -1, 1, 1;
%t (* First program *)
%t b[n_, k_, d_]:= If[n==k, 2, If[(k==d && n==d-2) || (n==d && k==d-2), -1, If[(k==n- 1 || k==n+1) && n<=d-1 && k<=d-1, -1, 0]]];
%t M[d_]:= Table[b[n, k, d], {n, d}, {k, d}];
%t p[x_, n_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
%t f = Table[p[x, n], {n, 0, 20}];
%t t[n_, k_]:= If[k==0, n!, Product[f[[j+1]], {j, n-1}]]/.x -> k+1;
%t T[n_, k_, m_]:= Round[t[n, m]/(t[k, m]*t[n-k, m])];
%t Table[T[n, k, 2], {n,0,15}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Jun 23 2021 *)
%t (* Second program *)
%t f[n_, x_]:= f[n,x]= If[n<2, (2-x)^n, (2-x)*LucasL[2*(n-1), Sqrt[-x]] ];
%t t[n_, k_]:= t[n,k]= If[k==0, n!, Product[f[j, x], {j, n-1}]]/.x -> (k+1);
%t T[n_, k_, m_]:= T[n,k,m]= Round[t[n,m]/(t[k,m]*t[n-k,m])];
%t Table[T[n, k, 2], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 23 2021 *)
%o (Sage)
%o @CachedFunction
%o def f(n,x): return (2-x)^n if (n<2) else 2*(2-x)*sum( ((n-1)/(2*n-j-2))*binomial(2*n-j-2, j)*(-x)^(n-j-1) for j in (0..n-1) )
%o def g(n,k): return factorial(n) if (k==0) else product( f(j, k+1) for j in (1..n-1) )
%o def T(n,k,m): return round( g(n,m)/(g(k,m)*g(n-k,m)) )
%o flatten([[T(n,k,2) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jun 23 2021
%Y Cf. A129862, A007318 (m=0), this sequence (m=2), A156609 (m=3), A156610 (m=4), A156612.
%K sign,tabl
%O 0,17
%A _Roger L. Bagula_, Feb 11 2009
%E Edited by _G. C. Greubel_, Jun 23 2021
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