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A284966 Triangle read by rows: coefficients of the scaled Lucas polynomials x^(n/2) L(n, sqrt(x)) for n >= 0. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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
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)).
Sequence in context: A138514 A286137 A320658 * A143540 A291336 A208664
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

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Last modified June 23 11:27 EDT 2024. Contains 373644 sequences. (Running on oeis4.)