

A092865


Nonzero elements in Klee's identity Sum[(1)^k binomial[n,k]binomial[n+k,m],{k,0,n}] == (1)^n binomial[n,mn].


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
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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, mn], {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



