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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, sorted by descending powers of x.

%I A284966 #45 Mar 09 2025 13:02:49
%S A284966 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 A284966 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 A284966 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 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.
%C A284966 For n >= 3, also the coefficients of the edge and vertex cover polynomials for the n-cycle graph C_n.
%C A284966 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 A284966 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 A284966 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 A284966 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>
%H A284966 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCoverPolynomial.html">Edge Cover Polynomial</a>
%H A284966 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasPolynomial.html">Lucas Polynomial</a>
%H A284966 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCoverPolynomial.html">Vertex Cover Polynomial</a>
%e A284966 First few polynomials are
%e A284966   2;
%e A284966   x;
%e A284966   2*x + x^2;
%e A284966   3*x^2 + x^3;
%e A284966   2*x^2 + 4*x^3 + x^4;
%e A284966   5*x^3 + 5*x^4 + x^5;
%e A284966   ...
%e A284966 giving
%e A284966   2;
%e A284966   0, 1;
%e A284966   0, 2, 1;
%e A284966   0, 0, 3, 1;
%e A284966   0, 0, 2, 4, 1;
%e A284966   0, 0, 0, 5, 5, 1;
%e A284966   ...
%p A284966 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 A284966 CoefficientList[Table[x^(n/2) LucasL[n, Sqrt[x]], {n, 12}], x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%t A284966 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 A284966 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 A284966 CoefficientList[LinearRecurrence[{x, x}, {x, x (2 + x)}, 15], x] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%Y A284966 Cf. A034807 (Lucas polynomials x^(n/2)*L(n, 1/sqrt(x))).
%Y A284966 Cf. A111125, A127677, A136481, A212634.
%K A284966 nonn,easy,tabl
%O A284966 0,1
%A A284966 _Eric W. Weisstein_, Apr 06 2017
%E A284966 First element T(n=0, k=0) and the example corrected by _Petros Hadjicostas_, Jan 27 2019
%E A284966 Name edited by _Petros Hadjicostas_, Jan 27 2019 and by _Stefano Spezia_, Mar 09 2025