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

1,4

Comments

Version of A053122 with row-ending 0's and differing signs.

Links

Table of n, a(n) for n=1..65.

Eric Weisstein's World of Mathematics, Path Graph

Eric Weisstein's World of Mathematics, Laplacian Polynomial

Formula

T(n,k) = (-1)^(k+1)*binomial[n+k,2*k+1]; 0 <= n <= k - Detlef Meya, Oct 09 2023

Example

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

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 *)

Prog

(Sage) # uses[riordan_square from A321620]
# Returns the triangle as a matrix.
riordan_square(-x/(1 - x)^2, 9) # Peter Luschny, Apr 03 2020

Crossrefs

Cf. A053122 (version lacking row-ending 0's and with differing signs).

Cf. A321620.

Sequence in context: A095884 A342240 A128908 * A300454 A155112 A368099

Adjacent sequences: A285069 A285070 A285071 * A285073 A285074 A285075

Keyword

sign,easy,tabl

Author

Eric W. Weisstein, Apr 09 2017

Status

approved