A284966 - OEIS

%I #40 Jan 05 2025 19:51:41

%S 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,

%T 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,

%U 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

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

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

%C 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

%H C. A. Charalambides, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/29-4/charalambides.pdf">Lucas numbers and polynomials of order k and the length of the longest circular success run</a>, The Fibonacci Quarterly, 29 (1991), 290-297.

%H W. O. J. Moser and M. Abramson, <a href="https://doi.org/10.1016/S0021-9800(69)80051-7">Enumeration of combinations with restricted differences and cospan</a>, J. Combin. Theory, 7 (1969), 162-170.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCoverPolynomial.html">Edge Cover Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasPolynomial.html">Lucas Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCoverPolynomial.html">Vertex Cover Polynomial</a>

%e First few polynomials are

%e 2;

%e x;

%e 2 x + x^2;

%e 3 x^2 + x^3;

%e 2 x^2 + 4 x^3 + x^4;

%e 5 x^3 + 5 x^4 + x^5;

%e ...

%e giving

%e 2;

%e 0, 1;

%e 0, 2, 1;

%e 0, 0, 3, 1;

%e 0, 0, 2, 4, 1;

%e 0, 0, 0, 5, 5, 1;

%e ...

%p 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

%t CoefficientList[Table[x^(n/2) LucasL[n, Sqrt[x]], {n, 12}], x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)

%t 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 *)

%t 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 *)

%t CoefficientList[LinearRecurrence[{x, x}, {x, x (2 + x)}, 15], x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)

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

%Y Cf. A111125, A127677, A136481, A212634.

%K nonn,easy,tabl,changed

%O 0,1

%A _Eric W. Weisstein_, Apr 06 2017

%E First element T(n=0, k=0) and the example corrected by _Petros Hadjicostas_, Jan 27 2019

%E Name edited by _Petros Hadjicostas_, Jan 27 2019