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A121872 Triangle T(n, k) = (k*ChebyshevU(n, (k+2)/2) + 2*ChebyshevT(n+1, (k+2)/2))/2.

%I A121872 #24 Sep 08 2022 08:45:27
%S A121872 5,13,41,34,153,436,89,571,2089,5741,233,2131,10009,33461,90481,610,
%T A121872 7953,47956,195025,620166,1663585,1597,29681,229771,1136689,4250681,
%U A121872 13097377,34988311,4181,110771,1100899,6625109,29134601,103115431,310957991,828931049
%N A121872 Triangle T(n, k) = (k*ChebyshevU(n, (k+2)/2) + 2*ChebyshevT(n+1, (k+2)/2))/2.
%H A121872 G. C. Greubel, <a href="/A121872/b121872.txt">Rows n = 1..100 of triangle, flattened</a>
%F A121872 T(n, m) = ((m+f(m))*(m+2 - f(m))^(n+2) - (m-f(m))*(m+2 + f(m))^(n+2))/( 2^(n+3)*f(m)), where f(m) = sqrt(m*(m+4)).
%F A121872 From _G. C. Greubel_, Oct 08 2019: (Start)
%F A121872 T(n, k) = (k*ChebyshevU(n, (k+2)/2) + 2*ChebyshevT(n+1, (k+2)/2))/2;
%F A121872 T(n, k) = (k*Fibonacci(n+2, m+2, -1) + Lucas(n+2, m+2, -1))/2, where Fibonacci(n, x, y) and Lucas(n, x, y) are the bi-variate Fibonacci an Lucas polynomials, respectively. (End)
%e A121872 Triangle begins as:
%e A121872     5;
%e A121872    13,   41;
%e A121872    34,  153,   436;
%e A121872    89,  571,  2089,  5741;
%e A121872   233, 2131, 10009, 33461, 90481;
%p A121872 seq(seq(simplify(( k*ChebyshevU(n, (k+2)/2) + 2*ChebyshevT(n+1, (k+2)/2) )/2), k=1..n), n=1..10); # _G. C. Greubel_, Oct 09 2019
%t A121872 f[k_]:= Sqrt[k*(k+4)]; T[n_, m_]:= T[n, m]= FullSimplify[((m+f[m])*(m+2 - f[m])^(n+2) - (m-f[m])*(m+2 + f[m])^(n+2))/(2^(n+3)*f[m])]; Table[T[n, m], {n,10}, {m,n}]//Flatten (* modified by _G. C. Greubel_, Oct 08 2019 *)
%t A121872 T[n_, k_]:= T[n, k]= (k*ChebyshevU[n, (k+2)/2] + 2*ChebyshevT[n+1, (k+ 2)/2])/2; Table[T[n, k], {n,10}, {k,n}]/Flatten (* _G. C. Greubel_, Oct 08 2019 *)
%o A121872 (PARI) T(n,k)= ( k*sin((n+1)*acos((k+2)/2))/sin(acos((k+2)/2)) + 2*cos((n+1)*acos((k+2)/2)) )/2;
%o A121872 for(n=1,10, for(k=1,n, print1(round(T(n,k)), ", "))) \\ _G. C. Greubel_, Oct 08 2019
%o A121872 (Magma)
%o A121872 T:= func< n,k | ( k*Sinh((n+1)*Argcosh((k+2)/2))/Sinh(Argcosh((k+2)/2)) + 2*Cosh((n+1)*Argcosh((k+2)/2)) )/2 >;
%o A121872 [Round(T(n,k)): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Oct 08 2019
%o A121872 (Sage)
%o A121872 [[( k*chebyshev_U(n,(k+2)/2) + 2*chebyshev_T(n+1, (k+2)/2) )/2 for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, Oct 08 2019
%Y A121872 Cf. A094954, A162997.
%Y A121872 Cf. A053117, A053120.
%K A121872 nonn,easy,tabl
%O A121872 1,1
%A A121872 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 09 2006
%E A121872 Major edit and new name, _G. C. Greubel_, Oct 08 2019