cp's OEIS Frontend

A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

%I A167925 #19 Sep 13 2023 02:14:25
%S A167925 0,1,1,1,2,3,0,2,6,12,-1,0,9,32,75,-1,-4,9,80,275,684,0,-8,0,192,1000,
%T A167925 3240,8232,1,-8,-27,448,3625,15336,47677,122368,1,0,-81,1024,13125,
%U A167925 72576,276115,835584,2158569,0,16,-162,2304,47500,343440,1599066,5705728,16953624,44010000
%N A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.
%H A167925 G. C. Greubel, <a href="/A167925/b167925.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A167925 T(n, k) = (-1)^(n+1) * [x^(n-1)]( 1/(1 + (k+1)*x + (k+1)*x^2) ). - _Francesco Daddi_, Aug 04 2011 (modified by _G. C. Greubel_, Sep 11 2023)
%F A167925 From _G. C. Greubel_, Sep 11 2023: (Start)
%F A167925 T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2).
%F A167925 T(n, 0) = A128834(n).
%F A167925 T(n, 1) = A009545(n) = A099087(n-1).
%F A167925 T(n, 2) = A057083(n-1).
%F A167925 T(n, 3) = A001787(n).
%F A167925 T(n, 4) = A030191(n-1).
%F A167925 T(n, 5) = A030192(n-1).
%F A167925 T(n, 6) = A030240(n-1).
%F A167925 T(n, 7) = A057084(n-1).
%F A167925 T(n, 8) = A057085(n).
%F A167925 T(n, 9) = A057086(n-1).
%F A167925 T(n, 10) = A190871(n).
%F A167925 T(n, 11) = A190873(n). (End)
%e A167925 Triangle begins as:
%e A167925    0;
%e A167925    1,  1;
%e A167925    1,  2,   3;
%e A167925    0,  2,   6,   12;
%e A167925   -1,  0,   9,   32,    75;
%e A167925   -1, -4,   9,   80,   275,   684;
%e A167925    0, -8,   0,  192,  1000,  3240,   8232;
%e A167925    1, -8, -27,  448,  3625, 15336,  47677, 122368;
%e A167925    1,  0, -81, 1024, 13125, 72576, 276115, 835584, 2158569;
%t A167925 (* First program *)
%t A167925 m[k_]= {{k,1}, {-1,1}};
%t A167925 v[0, k_]:= {0,1};
%t A167925 v[n_, k_]:= v[n, k]= m[k].v[n-1,k];
%t A167925 T[n_, k_]:= v[n, k][[1]];
%t A167925 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
%t A167925 (* Second program *)
%t A167925 A167925[n_, k_]:= (Sqrt[k+1])^(n-1)*ChebyshevU[n-1, Sqrt[k+1]/2];
%t A167925 Table[A167925[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 11 2023 *)
%o A167925 (Magma)
%o A167925 A167925:= func< n,k | Round((Sqrt(k+1))^(n-1)*Evaluate(ChebyshevSecond(n), Sqrt(k+1)/2)) >;
%o A167925 [A167925(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 11 2023
%o A167925 (SageMath)
%o A167925 def A167925(n,k): return (sqrt(k+1))^(n-1)*chebyshev_U(n-1, sqrt(k+1)/2)
%o A167925 flatten([[A167925(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 11 2023
%Y A167925 Cf. A009545, A030191, A030192, A030240, A057083, A057084, A057085, A057086, A099087, A128834, A190871, A190873.
%K A167925 sign,tabl
%O A167925 0,5
%A A167925 _Roger L. Bagula_, Nov 15 2009
%E A167925 Edited by _G. C. Greubel_, Sep 11 2023