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

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, 0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000

Offset: 0

Author

Roger L. Bagula, Nov 15 2009

Keywords

• sign
• tabl

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A009545, A030191, A030192, A030240, A057083, A057084, A057085, A057086, A099087, A128834, A190871, A190873.

Programs

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

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 *)

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

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, 1) = A009545(n) = A099087(n-1).

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)

Extensions

Edited by G. C. Greubel, Sep 11 2023