OFFSET
1,4
COMMENTS
Version of A053122 with row-ending 0's and differing signs.
LINKS
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
KEYWORD
AUTHOR
Eric W. Weisstein, Apr 09 2017
STATUS
approved