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