A285072 Triangle read by rows: coefficients of the Laplacian polynomial of the n-path graph P_n.
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, -10, 1, 0, -7, 56, -126, 120, -55, 12, -1, 0, -8, 84, -252, 330, -220, 78, -14, 1, 0, -9, 120, -462, 792, -715, 364, -105, 16, -1, 0, -10, 165, -792, 1716, -2002, 1365, -560, 136, -18, 1
Offset: 1
Examples
Table starts: 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 -10 1 0 -7 56 -126 120 -55 12 -1 0 -8 84 -252 330 -220 78 -14 1 0 -9 120 -462 792 -715 364 -105 16 -1 0
Links
- Eric Weisstein's World of Mathematics, Path Graph
- Eric Weisstein's World of Mathematics, Laplacian Polynomial
Programs
-
Maple
S := proc(n, k) option remember; if n <= k then 0 elif k = 0 then (-1)^n*n else S(n-1, k-1) - S(n-2, k) - 2*S(n-1, k) fi end: T := (n, k) -> (-1)^(n+1)*S(n, k): seq(seq(T(n, k), k=0..n), n=0..10); # Peter Luschny, Apr 03 2020
-
Mathematica
CoefficientList[Table[CharacteristicPolynomial[KirchhoffMatrix[PathGraph[Range[n]]], x], {n, 10}], x] // Flatten CoefficientList[LinearRecurrence[{2 - x, -1}, {-x, (-2 + x) x}, 10], x] // Flatten CoefficientList[Table[(-1)^(n + 1) x^(1/2) ChebyshevU[2 n - 1, -Sqrt[x]/2], {n, 10}], x] // Flatten 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[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 *) -
Sage
# uses[riordan_square from A321620] # Returns the triangle as a matrix. riordan_square(-x/(1 - x)^2, 9) # Peter Luschny, Apr 03 2020
Formula
T(n,k) = (-1)^(k+1)*binomial[n+k,2*k+1]; 0 <= n <= k - Detlef Meya, Oct 09 2023
Comments