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%I #5 Jun 25 2021 23:15:07
%S 1,1,1,1,-7,1,1,48,48,1,1,-329,2256,-329,1,1,2255,105985,105985,2255,
%T 1,1,-15456,4979040,-34127170,4979040,-15456,1,1,105937,233908896,
%U 10988845010,10988845010,233908896,105937,1,1,-726103,10988739073,-3538373981506,24252380937070,-3538373981506,10988739073,-726103,1
%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 = 8, read by rows.
%H G. C. Greubel, <a href="/A156602/b156602.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 = 8.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -7, 1;
%e 1, 48, 48, 1;
%e 1, -329, 2256, -329, 1;
%e 1, 2255, 105985, 105985, 2255, 1;
%e 1, -15456, 4979040, -34127170, 4979040, -15456, 1;
%e 1, 105937, 233908896, 10988845010, 10988845010, 233908896, 105937, 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, 8], {n,0,12}, {k,0,n}]//TableForm (* modified by _G. C. Greubel_, 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, 8], {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, 8) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 25 2021
%Y Cf. A007318 (m=0), A034801 (m=4), A156599 (m=5), A156600 (m=6), A156601 (m=7), this sequence (m=8), A156603.
%Y Cf. A053122.
%K sign,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 11 2009
%E Edited by _G. C. Greubel_, Jun 25 2021
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