cp's OEIS Frontend

A156601 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 = 7, read by rows.

%I A156601 #5 Jun 25 2021 23:14:57
%S A156601 1,1,1,1,-6,1,1,35,35,1,1,-204,1190,-204,1,1,1189,40426,40426,1189,1,
%T A156601 1,-6930,1373295,-8004348,1373295,-6930,1,1,40391,46651605,1584821667,
%U A156601 1584821667,46651605,40391,1,1,-235416,1584781276,-313786692648,1828884203718,-313786692648,1584781276,-235416,1
%N A156601 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 = 7, read by rows.
%H A156601 G. C. Greubel, <a href="/A156601/b156601.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A156601 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 = 7.
%e A156601 Triangle begins as:
%e A156601   1;
%e A156601   1,     1;
%e A156601   1,    -6,        1;
%e A156601   1,    35,       35,          1;
%e A156601   1,  -204,     1190,       -204,          1;
%e A156601   1,  1189,    40426,      40426,       1189,        1;
%e A156601   1, -6930,  1373295,   -8004348,    1373295,    -6930,     1;
%e A156601   1, 40391, 46651605, 1584821667, 1584821667, 46651605, 40391, 1;
%t A156601 (* First program *)
%t A156601 b[n_, k_]:= If[k==n, 2, If[k==n-1 || k==n+1, -1, 0]];
%t A156601 M[d_]:= Table[b[n, k], {n,d}, {k,d}];
%t A156601 p[x_, n_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
%t A156601 f= Table[p[x, n], {n,0,20}];
%t A156601 t[n_, k_]:= If[k==0, n!, Product[f[[j]], {j, n}]/.x->(k+1)];
%t A156601 T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];
%t A156601 Table[T[n, k, 7], {n,0,12}, {k,0,n}]//TableForm (* modified by _G. C. Greubel_, Jun 25 2021 *)
%t A156601 (* Second program *)
%t A156601 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 A156601 T[n_, k_, m_]:= T[n, k, m]= t[n, m]/(t[k, m]*t[n-k, m]);
%t A156601 Table[T[n, k, 7], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 25 2021 *)
%o A156601 (Sage)
%o A156601 @CachedFunction
%o A156601 def t(n, k):
%o A156601     if (n==0): return 1
%o A156601     elif (k==0): return factorial(n-1)
%o A156601     else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
%o A156601 def T(n,k,m): return t(n,m)/(t(k,m)*t(n-k,m))
%o A156601 flatten([[T(n, k, 7) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 25 2021
%Y A156601 Cf. A007318 (m=0), A034801 (m=4), A156599 (m=5), A156600 (m=6), this sequence (m=7), A156602 (m=8), A156603.
%Y A156601 Cf. A053122.
%K A156601 sign,tabl
%O A156601 0,5
%A A156601 _Roger L. Bagula_, Feb 11 2009
%E A156601 Edited by _G. C. Greubel_, Jun 25 2021