OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
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)
From G. C. Greubel, Sep 11 2023: (Start)
T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2).
T(n, 0) = A128834(n).
T(n, 2) = A057083(n-1).
T(n, 3) = A001787(n).
T(n, 4) = A030191(n-1).
T(n, 5) = A030192(n-1).
T(n, 6) = A030240(n-1).
T(n, 7) = A057084(n-1).
T(n, 8) = A057085(n).
T(n, 9) = A057086(n-1).
T(n, 10) = A190871(n).
T(n, 11) = A190873(n). (End)
EXAMPLE
Triangle begins as:
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, 3240, 8232;
1, -8, -27, 448, 3625, 15336, 47677, 122368;
1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569;
MATHEMATICA
(* First program *)
m[k_]= {{k, 1}, {-1, 1}};
v[0, k_]:= {0, 1};
v[n_, k_]:= v[n, k]= m[k].v[n-1, k];
T[n_, k_]:= v[n, k][[1]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
(* Second program *)
A167925[n_, k_]:= (Sqrt[k+1])^(n-1)*ChebyshevU[n-1, Sqrt[k+1]/2];
Table[A167925[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 11 2023 *)
PROG
(Magma)
A167925:= func< n, k | Round((Sqrt(k+1))^(n-1)*Evaluate(ChebyshevSecond(n), Sqrt(k+1)/2)) >;
[A167925(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 11 2023
(SageMath)
def A167925(n, k): return (sqrt(k+1))^(n-1)*chebyshev_U(n-1, sqrt(k+1)/2)
flatten([[A167925(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 11 2023
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Nov 15 2009
EXTENSIONS
Edited by G. C. Greubel, Sep 11 2023
STATUS
approved