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%I #7 Jun 25 2021 23:15:21
%S 1,1,1,1,1,2,1,1,0,6,1,1,-1,0,24,1,1,-2,0,0,120,1,1,-3,-6,0,0,720,1,1,
%T -4,-24,24,0,0,5040,1,1,-5,-60,504,120,0,0,40320,1,1,-6,-120,3360,
%U 27720,-720,0,0,362880,1,1,-7,-210,13800,702240,-3991680,-5040,0,0,3628800
%N Square array 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 T(n, 0) = n!, read by antidiagonals.
%H G. C. Greubel, <a href="/A156603/b156603.txt">Antidiagonal rows n = 0..50, flattened</a>
%F 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 T(n, 0) = n! (square array).
%F T(n, k) = Product_{j=0..n-1} (-1)^j*ChebyshevU(j, (k-1)/2) with T(n, 0) = n! for n >= 1, and T(0, k) = 1 (square array). - _G. C. Greubel_, Jun 25 2021
%e Square array begins as:
%e 1, 1, 1, 1, 1, 1, 1 ...;
%e 1, 1, 1, 1, 1, 1, 1 ...;
%e 2, 0, -1, -2, -3, -4, -5 ...;
%e 6, 0, 0, -6, -24, -60, -120 ...;
%e 24, 0, 0, 24, 504, 3360, 13800 ...;
%e 120, 0, 0, 120, 27720, 702240, 7603800 ...;
%e 720, 0, 0, -720, -3991680, -547747200, -20074032000 ...;
%e Antidiagonal rows begin as:
%e 1;
%e 1, 1;
%e 1, 1, 2;
%e 1, 1, 0, 6;
%e 1, 1, -1, 0, 24;
%e 1, 1, -2, 0, 0, 120;
%e 1, 1, -3, -6, 0, 0, 720;
%e 1, 1, -4, -24, 24, 0, 0, 5040;
%e 1, 1, -5, -60, 504, 120, 0, 0, 40320;
%e 1, 1, -6, -120, 3360, 27720, -720, 0, 0, 362880;
%e 1, 1, -7, -210, 13800, 702240, -3991680, -5040, 0, 0, 3628800;
%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 Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 25 2021 *)
%t (* Second program *)
%t T[n_, k_]:= If[n==0, 1, If[k==0, n!, Product[(-1)^j*Simplify[ChebyshevU[j, x/2-1]], {j, 0, n-1}]/.x->(k+1)]];
%t Table[T[k, n-k], {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)
%o else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
%o flatten([[T(k, n-k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jun 25 2021
%Y Cf. A007318, A034801, A053122, A156599, A156600, A156601, A156602.
%K sign,tabl
%O 0,6
%A _Roger L. Bagula_, Feb 11 2009
%E Edited by _G. C. Greubel_, Jun 25 2021
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