cp's OEIS Frontend

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].

%I A092865 #39 Feb 16 2025 08:32:53
%S A092865 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,
%T A092865 20,-21,8,-1,-1,15,-35,28,-9,1,6,-35,56,-36,10,-1,1,-21,70,-84,45,-11,
%U A092865 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
%N 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].
%C A092865 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
%C A092865 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
%H A092865 H.-H. Chern, H.-K. Hwang, T.-H. Tsai, <a href="http://arxiv.org/abs/1406.0614">Random unfriendly seating arrangement in a dining table</a>, arXiv preprint arXiv:1406.0614 [math.PR], 2014
%H A092865 T. Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>
%H A092865 P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
%H A092865 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KleesIdentity.html">Klee's Identity</a>
%F A092865 G.f.: 1/(1+y*x+y*x^2). - _Philippe Deléham_, Feb 08 2012
%e A092865 1;
%e A092865 -1;
%e A092865 -1, 1;
%e A092865 2, -1;
%e A092865 1, -3, 1;
%e A092865 -3, 4, -1;
%e A092865 -1, 6, -5, 1;
%e A092865 4, -10, 6, -1;
%e A092865 Triangle (0, 1, -1, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, ...) begins:
%e A092865 1
%e A092865 0, -1
%e A092865 0, -1, 1
%e A092865 0, 0, 2, -1
%e A092865 0, 0, 1, -3, 1
%e A092865 0, 0, 0, -3, 4, -1
%e A092865 0, 0, 0, -1, 6, -5, 1 ... - _Philippe Deléham_, Dec 26 2011
%t A092865 Flatten[Table[(-1)^n Binomial[n, m-n], {m, 0, 20}, {n, Ceiling[m/2], m}]]
%Y A092865 All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - _N. J. A. Sloane_, May 29 2011
%K A092865 sign,tabf
%O A092865 0,5
%A A092865 _Eric W. Weisstein_, Mar 07 2004