A104562 Inverse of the Motzkin triangle A064189.

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, -12, 1

Offset: 0

Author

Paul Barry, Mar 15 2005

Keywords

• easy
• sign
• tabl

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

Examples

			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)
		

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.

Crossrefs

Apart from signs identical to A101950.

Cf. A125090.

Programs

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 *)

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

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

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