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A092865 Nonzero elements in Klee's identity Sum[(-1)^k binomial[n,k]binomial[n+k,m],{k,0,n}] == (-1)^n binomial[n,m-n]. 9
1, -1, -1, 1, 2, -1, 1, -3, 1, -3, 4, -1, -1, 6, -5, 1, 4, -10, 6, -1, 1, -10, 15, -7, 1, -5, 20, -21, 8, -1, -1, 15, -35, 28, -9, 1, 6, -35, 56, -36, 10, -1, 1, -21, 70, -84, 45, -11, 1, -7, 56, -126, 120, -55, 12, -1, -1, 28, -126, 210, -165, 66, -13, 1, 8, -84, 252, -330, 220, -78, 14, -1, 1, -36, 210, -462, 495 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Triangle, with zeros omitted, given by (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 26 2011

Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014

LINKS

Table of n, a(n) for n=0..76.

H.-H. Chern, H.-K. Hwang, T.-H. Tsai, Random unfriendly seating arrangement in a dining table, arXiv preprint arXiv:1406.0614 [math.PR], 2014

T. Copeland, Addendum to Elliptic Lie Triad

P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.

Eric Weisstein's World of Mathematics, Klee's Identity

FORMULA

G.f.: 1/(1+y*x+y*x^2). - Philippe Deléham, Feb 08 2012

EXAMPLE

1;

-1;

-1, 1;

2, -1;

1, -3, 1;

-3, 4, -1;

-1, 6, -5, 1;

4, -10, 6, -1;

Triangle (0, 1, -1, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, ...) begins:

1

0, -1

0, -1, 1

0, 0, 2, -1

0, 0, 1, -3, 1

0, 0, 0, -3, 4, -1

0, 0, 0, -1, 6, -5, 1 ... - Philippe Deléham, Dec 26 2011

MATHEMATICA

Flatten[Table[(-1)^n Binomial[n, m-n], {m, 0, 20}, {n, Ceiling[m/2], m}]]

CROSSREFS

All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - N. J. A. Sloane, May 29 2011

Sequence in context: A308399 A287601 A035667 * A098925 A102426 A052920

Adjacent sequences:  A092862 A092863 A092864 * A092866 A092867 A092868

KEYWORD

sign,tabf

AUTHOR

Eric W. Weisstein, Mar 07 2004

STATUS

approved

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Last modified December 8 12:25 EST 2019. Contains 329862 sequences. (Running on oeis4.)