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A285072 Triangle read by rows: coefficients of the Laplacian polynomial of the n-path graph P_n.

%I A285072 #27 Feb 16 2025 08:33:43
%S A285072 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 A285072 -10,1,0,-7,56,-126,120,-55,12,-1,0,-8,84,-252,330,-220,78,-14,1,0,-9,
%U A285072 120,-462,792,-715,364,-105,16,-1,0,-10,165,-792,1716,-2002,1365,-560,136,-18,1
%N A285072 Triangle read by rows: coefficients of the Laplacian polynomial of the n-path graph P_n.
%C A285072 Version of A053122 with row-ending 0's and differing signs.
%H A285072 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H A285072 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaplacianPolynomial.html">Laplacian Polynomial</a>
%F A285072 T(n,k) = (-1)^(k+1)*binomial[n+k,2*k+1]; 0 <= n <= k - _Detlef Meya_, Oct 09 2023
%e A285072 Table starts:
%e A285072   0
%e A285072  -1    0
%e A285072  -2    1    0
%e A285072  -3    4   -1     0
%e A285072  -4   10   -6     1     0
%e A285072  -5   20  -21     8    -1    0
%e A285072  -6   35  -56    36   -10    1     0
%e A285072  -7   56  -126  120   -55   12    -1   0
%e A285072  -8   84  -252  330  -220   78   -14   1   0
%e A285072  -9  120  -462  792  -715  364  -105  16  -1   0
%p A285072 S := proc(n, k) option remember;
%p A285072 if n <= k then 0 elif k = 0 then (-1)^n*n
%p A285072 else S(n-1, k-1) - S(n-2, k) - 2*S(n-1, k) fi end:
%p A285072 T := (n, k) -> (-1)^(n+1)*S(n, k):
%p A285072 seq(seq(T(n, k), k=0..n), n=0..10); # _Peter Luschny_, Apr 03 2020
%t A285072 CoefficientList[Table[CharacteristicPolynomial[KirchhoffMatrix[PathGraph[Range[n]]], x], {n, 10}], x] // Flatten
%t A285072 CoefficientList[LinearRecurrence[{2 - x, -1}, {-x, (-2 + x) x}, 10], x] // Flatten
%t A285072 CoefficientList[Table[(-1)^(n + 1) x^(1/2) ChebyshevU[2 n - 1, -Sqrt[x]/2], {n, 10}], x] // Flatten
%t A285072 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 A285072 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 A285072 (Sage) # uses[riordan_square from A321620]
%o A285072 # Returns the triangle as a matrix.
%o A285072 riordan_square(-x/(1 - x)^2, 9) # _Peter Luschny_, Apr 03 2020
%Y A285072 Cf. A053122 (version lacking row-ending 0's and with differing signs).
%Y A285072 Cf. A321620.
%K A285072 sign,easy,tabl
%O A285072 1,4
%A A285072 _Eric W. Weisstein_, Apr 09 2017