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

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

0,1

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

Links

Table of n, a(n) for n=0..89.

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

Example

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;
...

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 *)

Crossrefs

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

Cf. A111125, A127677, A136481, A212634.

Sequence in context: A138514 A286137 A320658 * A143540 A291336 A208664

Adjacent sequences: A284963 A284964 A284965 * A284967 A284968 A284969

Keyword

nonn,easy,tabl

Author

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

Status

approved