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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. 5

%I #5 Jun 25 2021 23:14:57

%S 1,1,1,1,-6,1,1,35,35,1,1,-204,1190,-204,1,1,1189,40426,40426,1189,1,

%T 1,-6930,1373295,-8004348,1373295,-6930,1,1,40391,46651605,1584821667,

%U 1584821667,46651605,40391,1,1,-235416,1584781276,-313786692648,1828884203718,-313786692648,1584781276,-235416,1

%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 = 7, read by rows.

%H G. C. Greubel, <a href="/A156601/b156601.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 = 7.

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, -6, 1;

%e 1, 35, 35, 1;

%e 1, -204, 1190, -204, 1;

%e 1, 1189, 40426, 40426, 1189, 1;

%e 1, -6930, 1373295, -8004348, 1373295, -6930, 1;

%e 1, 40391, 46651605, 1584821667, 1584821667, 46651605, 40391, 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, 7], {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, 7], {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, 7) 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), this sequence (m=7), A156602 (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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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)