OFFSET
0,5
COMMENTS
Or, triangle read by rows: T(0, 0) = 1; for n >= 1 T(n, k) is the coefficient of x^k in the monic characteristic polynomial of the n X n tridiagonal matrix with 1's on the main, sub- and superdiagonal (0 <= k <= n). The characteristic polynomial has a root 1 + 2*cos(Pi/(n + 1)). - Gary W. Adamson, Nov 19 2006
Row sums have g.f. 1/(1 + x^2); diagonal sums are (-1)^n. Riordan array (1/(1 + x + x^2), x/(1 + x + x^2)).
Or, triangle read by rows in which row n gives coefficients of characteristic polynomial of the n X n tridiagonal matrix with 1's on the main diagonal and -1's on the two adjacent diagonals. For example: M(3) = {{1, -1, 0}, {-1, 1, -1}, {0, -1, 1}}. - Roger L. Bagula, Mar 15 2008
Subtriangle of the triangle given by [0,-1,1,-1,0,0,0,0,0,0,0,...) DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 27 2010
Triangle of coefficients of Chebyshev's S(n, x-1) polynomials (exponents of x in increasing order). - Philippe Deléham, Feb 19 2012
REFERENCES
Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
LINKS
Paul Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, Vol. 15 2012, #12.8.2.
Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker, and David G. L. Wang, Root geometry of polynomial sequences. II: Type (1,0), J. Math. Anal. Appl. 441, No. 2, 499-528 (2016).
A. Luzón, D. Merlini, M. A. Morón, and R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.
FORMULA
T(n, k) = Sum_{j=0..n} (-1)^(k-j)*(-1)^((n-j)/2) C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2.
T(n,k) = (-1)^(n-k)*A101950(n,k). - Philippe Deléham, Feb 19 2012
T(n,k) = T(n-1,k-1) - T(n-1,k) - T(n-2,l). - Philippe Deléham, Feb 19 2012
G.f.: 1/(1+x+x^2-y*x). - Philippe Deléham, Feb 19 2012
T(n, k) = (-1)^(n - k)*C(n, k)*hypergeom([(k - n)/2, (k - n + 1)/2], [-n], 4) for n >= 1. - Peter Luschny, Apr 25 2016
EXAMPLE
Triangle starts:
[0] 1;
[1] -1, 1;
[2] 0, -2, 1;
[3] 1, 1, -3, 1;
[4] -1, 2, 3, -4, 1;
[5] 0, -4, 2, 6, -5, 1;
[6] 1, 2, -9, 0, 10, -6, 1;
[7] -1, 3, 9, -15, -5, 15, -7, 1;
[8] 0, -6, 3, 24, -20, -14, 21, -8, 1;
[9] 1, 3, -18, -6, 49, -21, -28, 28, -9, 1.
...
From Philippe Deléham, Jan 27 2010: (Start)
Triangle [0,-1,1,-1,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] begins:
1;
0, 1;
0, -1, 1;
0, 0, -2, 1;
0, 1, 1, -3, 1;
0, -1, 2, 3, -4, 1;
... (End)
MAPLE
with(linalg): m:=proc(i, j) if abs(i-j)<=1 then 1 else 0 fi end: T:=(n, k)->coeff(charpoly(matrix(n, n, m), x), x, k): 1; for n from 1 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
# Alternatively:
T := (n, k) -> `if`(n=0, 1, (-1)^(n-k)*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], 4)): seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Apr 25 2016
MATHEMATICA
nmax = 12;
M[n_, k_] := Binomial[n, k] Hypergeometric2F1[(k-n)/2, (k-n+1)/2, k+2, 4];
invM = Inverse@Table[M[n, k], {n, 0, nmax}, {k, 0, nmax}];
T[n_, k_] := invM[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 15 2023 *)
PROG
(Sage)
@CachedFunction
def A104562(n, k):
if n< 0: return 0
if n==0: return 1 if k == 0 else 0
for n in (0..9): [A104562(n, k) for k in (0..n)] # Peter Luschny, Nov 20 2012
(Sage) # Alternatively as coefficients of polynomials:
def S(n, x):
if n==0: return 1
if n==1: return x-1
return (x-1)*S(n-1, x)-S(n-2, x)
for n in (0..7): print(S(n, x).list()) # Peter Luschny, Jun 23 2015
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Mar 15 2005
EXTENSIONS
Edited by N. J. A. Sloane, Apr 10 2008
Typo correction in the Roger L. Bagula comment and Mathematica section by Wolfdieter Lang, Nov 22 2011
STATUS
approved