This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A156610 #9 Jun 25 2021 03:38:49
%S A156610 1,1,1,1,-3,1,1,9,9,1,1,-21,63,-21,1,1,54,378,378,54,1,1,-141,2538,
%T A156610 -5922,2538,-141,1,1,369,17343,104058,104058,17343,369,1,1,-966,
%U A156610 118818,-1861482,4786668,-1861482,118818,-966,1,1,2529,814338,33387858,224175618,224175618,33387858,814338,2529,1
%N A156610 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 = 4, read by rows.
%C A156610 Cartan_Dn refers to a Cartan matrix of type D_n. - _N. J. A. Sloane_, Jun 25 2021
%H A156610 G. C. Greubel, <a href="/A156610/b156610.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A156610 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 = 4.
%F A156610 T(n, 1) = T(n, n-1) = [n==1] - 3*A219233(n-2)*[n >= 2]. - _G. C. Greubel_, Jun 24 2021
%e A156610 Triangle begins:
%e A156610 1;
%e A156610 1, 1;
%e A156610 1, -3, 1;
%e A156610 1, 9, 9, 1;
%e A156610 1, -21, 63, -21, 1;
%e A156610 1, 54, 378, 378, 54, 1;
%e A156610 1, -141, 2538, -5922, 2538, -141, 1;
%e A156610 1, 369, 17343, 104058, 104058, 17343, 369, 1;
%e A156610 1, -966, 118818, -1861482, 4786668, -1861482, 118818, -966, 1;
%e A156610 1, 2529, 814338, 33387858, 224175618, 224175618, 33387858, 814338, 2529, 1;
%t A156610 (* First program *)
%t A156610 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 A156610 M[d_]:= Table[b[n, k, d], {n, d}, {k, d}];
%t A156610 p[x_, n_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
%t A156610 f = Table[p[x, n], {n, 0, 20}];
%t A156610 t[n_, k_]:= If[k==0, n!, Product[f[[j+1]], {j, n-1}]]/.x -> k+1;
%t A156610 T[n_, k_, m_]:= Round[t[n, m]/(t[k, m]*t[n-k, m])];
%t A156610 Table[T[n, k, 4], {n,0,15}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Jun 24 2021 *)
%t A156610 (* Second program *)
%t A156610 f[n_, x_]:= f[n,x]= If[n<2, (2-x)^n, (2-x)*LucasL[2*(n-1), Sqrt[-x]] ];
%t A156610 t[n_, k_]:= t[n,k]= If[k==0, n!, Product[f[j, x], {j, n-1}]]/.x -> (k+1);
%t A156610 T[n_, k_, m_]:= T[n,k,m]= Round[t[n,m]/(t[k,m]*t[n-k,m])];
%t A156610 Table[T[n, k, 4], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 24 2021 *)
%o A156610 (Sage)
%o A156610 @CachedFunction
%o A156610 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 A156610 def g(n,k): return factorial(n) if (k==0) else product( f(j, k+1) for j in (1..n-1) )
%o A156610 def T(n,k,m): return round( g(n,m)/(g(k,m)*g(n-k,m)) )
%o A156610 flatten([[T(n,k,4) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jun 24 2021
%Y A156610 Cf. A129862, A007318 (m=0), A156608 (m=2), A156609 (m=3), this sequence (m=4), A156612.
%K A156610 sign,tabl
%O A156610 0,5
%A A156610 _Roger L. Bagula_, Feb 11 2009
%E A156610 Definition corrected and edited by _G. C. Greubel_, Jun 24 2021