A284966 Triangle read by rows: coefficients of the scaled Lucas polynomials x^(n/2)*L(n, sqrt(x)) for n >= 0, sorted by descending powers of x.

2, 1, 0, 2, 1, 0, 0, 3, 1, 0, 0, 2, 4, 1, 0, 0, 0, 5, 5, 1, 0, 0, 0, 2, 9, 6, 1, 0, 0, 0, 0, 7, 14, 7, 1, 0, 0, 0, 0, 2, 16, 20, 8, 1, 0, 0, 0, 0, 0, 9, 30, 27, 9, 1, 0, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 0, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1, 0, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1

Offset: 0

Author

Eric W. Weisstein, Apr 06 2017

Keywords

• nonn
• easy
• tabl

Comments

For n >= 3, also the coefficients of the edge and vertex cover polynomials for the n-cycle graph C_n.

For more information on how this triangular array is related to the work of Charalambides (1991) and Moser and Abramson (1969), see the comments for triangular array A212634 (which contains additional formulas). The coefficients of these polynomials are given by formula (2.1), p. 291, in Charalambides (1991), where an obvious typo in the index of the summation must be corrected (floor(n/K) -> floor(n/K) - 1). - Petros Hadjicostas, Jan 27 2019

Examples

			First few polynomials are
  2;
  x;
  2*x + x^2;
  3*x^2 + x^3;
  2*x^2 + 4*x^3 + x^4;
  5*x^3 + 5*x^4 + x^5;
  ...
giving
  2;
  0, 1;
  0, 2, 1;
  0, 0, 3, 1;
  0, 0, 2, 4, 1;
  0, 0, 0, 5, 5, 1;
  ...
		

Links

• C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
• W. O. J. Moser and M. Abramson, Enumeration of combinations with restricted differences and cospan, J. Combin. Theory, 7 (1969), 162-170.
• Eric Weisstein's World of Mathematics, Cycle Graph
• Eric Weisstein's World of Mathematics, Edge Cover Polynomial
• Eric Weisstein's World of Mathematics, Lucas Polynomial
• Eric Weisstein's World of Mathematics, Vertex Cover Polynomial

Crossrefs

Cf. A034807 (Lucas polynomials x^(n/2)*L(n, 1/sqrt(x))).

Cf. A111125, A127677, A136481, A212634.

Programs

Maple

L := proc (n, K, x) -1 + sum((-1)^j*n*binomial(n - j*K, j)*x^j*(x+1)^(n - j*(K+1))/(n - j*K), j = 0 .. floor(n/(K + 1))) end proc; for i to 30 do expand(L(i, 2, x)) end do; # gives the g.f. of row n for 1 <= n <= 30. - Petros Hadjicostas, Jan 27 2019

Mathematica

CoefficientList[Table[x^(n/2) LucasL[n, Sqrt[x]], {n, 12}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[2 x^n (-1/x)^(n/2) ChebyshevT[n, 1/(2 Sqrt[-1/x])], {n, 12}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[FunctionExpand[2 (-(1/x))^(n/2) x^n Cos[n ArcSec[2 Sqrt[-(1/x)]]]], {n, 15}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[LinearRecurrence[{x, x}, {x, x (2 + x)}, 15], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)

Extensions

First element T(n=0, k=0) and the example corrected by Petros Hadjicostas, Jan 27 2019

Name edited by Petros Hadjicostas, Jan 27 2019 and by Stefano Spezia, Mar 09 2025