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A092865
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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].
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9
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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
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0,5
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COMMENTS
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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
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LINKS
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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
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FORMULA
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G.f.: 1/(1+y*x+y*x^2). - Philippe Deléham, Feb 08 2012
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EXAMPLE
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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
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MATHEMATICA
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Flatten[Table[(-1)^n Binomial[n, m-n], {m, 0, 20}, {n, Ceiling[m/2], m}]]
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CROSSREFS
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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
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KEYWORD
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sign,tabf
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AUTHOR
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Eric W. Weisstein, Mar 07 2004
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STATUS
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approved
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