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A156599 Triangle T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where t(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and m = 5, read by rows.

%I A156599 #11 Jun 25 2021 23:14:28
%S A156599 1,1,1,1,-4,1,1,15,15,1,1,-56,210,-56,1,1,209,2926,2926,209,1,1,-780,
%T A156599 40755,-152152,40755,-780,1,1,2911,567645,7909187,7909187,567645,2911,
%U A156599 1,1,-10864,7906276,-411126352,1534382278,-411126352,7906276,-10864,1,1,40545,110120220,21370664028,297662820390,297662820390,21370664028,110120220,40545,1
%N A156599 Triangle T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where t(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and m = 5, read by rows.
%H A156599 G. C. Greubel, <a href="/A156599/b156599.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A156599 T(n, k, m) = t(n, m)/(t(k, m)*t(n-k, m)), where t(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and m = 5.
%e A156599 Triangle begins:
%e A156599   1;
%e A156599   1,    1;
%e A156599   1,   -4,      1;
%e A156599   1,   15,     15,       1;
%e A156599   1,  -56,    210,     -56,       1;
%e A156599   1,  209,   2926,    2926,     209,      1;
%e A156599   1, -780,  40755, -152152,   40755,   -780,    1;
%e A156599   1, 2911, 567645, 7909187, 7909187, 567645, 2911, 1;
%t A156599 (* First program *)
%t A156599 b[n_, k_]:= If[k==n, 2, If[k==n-1 || k==n+1, -1, 0]];
%t A156599 M[d_]:= Table[b[n, k], {n,d}, {k,d}];
%t A156599 p[x_, n_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
%t A156599 f= Table[p[x, n], {n,0,20}];
%t A156599 t[n_, k_]:= If[k==0, n!, Product[f[[j]], {j, n}]/.x->(k+1)];
%t A156599 T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];
%t A156599 Table[T[n, k, 5], {n,0,12}, {k,0,n}]//TableForm (* modified by _G. C. Greubel_, May 23 2019; Jun 25 2021 *)
%t A156599 (* Second program *)
%t A156599 t[n_, k_]:= t[n, k]= If[n==0, 1, If[k==0, (n-1)!, Product[(-1)^j*Simplify[ChebyshevU[j, x/2 - 1]], {j,0,n-1}]/.x->(k+1)]];
%t A156599 T[n_, k_, m_]:= T[n, k, m]= t[n, m]/(t[k, m]*t[n-k, m]);
%t A156599 Table[T[n, k, 5], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 25 2021 *)
%o A156599 (Sage)
%o A156599 @CachedFunction
%o A156599 def t(n, k):
%o A156599     if (n==0): return 1
%o A156599     elif (k==0): return factorial(n-1)
%o A156599     else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
%o A156599 def T(n,k,m): return t(n,m)/(t(k,m)*t(n-k,m))
%o A156599 flatten([[T(n, k, 5) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 25 2021
%Y A156599 Cf. A007318 (m=0), A034801 (m=4), this sequence (m=5), A156600 (m=6), A156601 (m=7), A156602 (m=8), A156603.
%Y A156599 Cf. A053122.
%K A156599 sign,tabl,less
%O A156599 0,5
%A A156599 _Roger L. Bagula_, Feb 11 2009
%E A156599 Edited by _G. C. Greubel_, May 23 2019; Jun 25 2021