|
| |
|
|
A104562
|
|
Inverse of the Motzkin triangle A064189.
|
|
13
|
|
|
|
1, -1, 1, 0, -2, 1, 1, 1, -3, 1, -1, 2, 3, -4, 1, 0, -4, 2, 6, -5, 1, 1, 2, -9, 0, 10, -6, 1, -1, 3, 9, -15, -5, 15, -7, 1, 0, -6, 3, 24, -20, -14, 21, -8, 1, 1, 3, -18, -6, 49, -21, -28, 28, -9, 1, -1, 4, 18, -36, -35, 84, -14, -48, 36, -10, 1, 0, -8, 4, 60, -50, -98, 126, 6, -75, 45, -11, 1, 1, 4, -30, -20, 145, -36, -210, 168, 45, -110, 55
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
Table of n, a(n) for n=0..88.
Paul Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, Vol. 15 2012, #12.8.2.
Gross, Jonathan L. ; Mansour, Toufik; Tucker, Thomas W.; Wang, David G. L. 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, 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
A104562*A007318 = A049310 as infinite lower triangular matrices. - 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
|
|
|
PROG
|
(Sage)
@CachedFunction
def A104562(n, k):
if n< 0: return 0
if n==0: return 1 if k == 0 else 0
return A104562(n-1, k-1)-A104562(n-2, k)-A104562(n-1, k)
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
|
Apart from signs identical to A101950.
Cf. A125090.
Sequence in context: A333381 A124094 A101950 * A164306 A309931 A309939
Adjacent sequences: A104559 A104560 A104561 * A104563 A104564 A104565
|
|
|
KEYWORD
|
easy,sign,tabl
|
|
|
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
|
| |
|
|
