%I #26 Oct 09 2023 15:19:27
%S 0,-1,0,-2,1,0,-3,4,-1,0,-4,10,-6,1,0,-5,20,-21,8,-1,0,-6,35,-56,36,
%T -10,1,0,-7,56,-126,120,-55,12,-1,0,-8,84,-252,330,-220,78,-14,1,0,-9,
%U 120,-462,792,-715,364,-105,16,-1,0,-10,165,-792,1716,-2002,1365,-560,136,-18,1
%N Triangle read by rows: coefficients of the Laplacian polynomial of the n-path graph P_n.
%C Version of A053122 with row-ending 0's and differing signs.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaplacianPolynomial.html">Laplacian Polynomial</a>
%F T(n,k) = (-1)^(k+1)*binomial[n+k,2*k+1]; 0 <= n <= k - _Detlef Meya_, Oct 09 2023
%e Table starts:
%e 0
%e -1 0
%e -2 1 0
%e -3 4 -1 0
%e -4 10 -6 1 0
%e -5 20 -21 8 -1 0
%e -6 35 -56 36 -10 1 0
%e -7 56 -126 120 -55 12 -1 0
%e -8 84 -252 330 -220 78 -14 1 0
%e -9 120 -462 792 -715 364 -105 16 -1 0
%p S := proc(n, k) option remember;
%p if n <= k then 0 elif k = 0 then (-1)^n*n
%p else S(n-1, k-1) - S(n-2, k) - 2*S(n-1, k) fi end:
%p T := (n, k) -> (-1)^(n+1)*S(n, k):
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Peter Luschny_, Apr 03 2020
%t CoefficientList[Table[CharacteristicPolynomial[KirchhoffMatrix[PathGraph[Range[n]]], x], {n, 10}], x] // Flatten
%t CoefficientList[LinearRecurrence[{2 - x, -1}, {-x, (-2 + x) x}, 10], x] // Flatten
%t CoefficientList[Table[(-1)^(n + 1) x^(1/2) ChebyshevU[2 n - 1, -Sqrt[x]/2], {n, 10}], x] // Flatten
%t CoefficientList[Table[(2^-n ((2 - Sqrt[-4 + x] Sqrt[x] - x)^n - (2 + Sqrt[-4 + x] Sqrt[x] - x)^n))/Sqrt[(-4 + x)/x], {n, 10}] // Expand // FullSimplify, x] // Flatten
%t T[n_,k_]:=(-1)^(k+1)*Binomial[n+k,2*k+1];Flatten[Table[T[n,k],{n,0,10},{k,0,n}]] (* _Detlef Meya_, Oct 09 2023 *)
%o (Sage) # uses[riordan_square from A321620]
%o # Returns the triangle as a matrix.
%o riordan_square(-x/(1 - x)^2, 9) # _Peter Luschny_, Apr 03 2020
%Y Cf. A053122 (version lacking row-ending 0's and with differing signs).
%Y Cf. A321620.
%K sign,easy,tabl
%O 1,4
%A _Eric W. Weisstein_, Apr 09 2017
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