A263848 - OEIS

%I #18 Dec 14 2015 12:02:01

%S 1,1,-1,1,0,-1,1,-1,1,1,0,0,-1,2,0,-1,1,2,-1,0,1,1,-1,1,-1,1,0,0,0,-1,

%T 3,0,0,-1,1,5,0,-1,0,1,3,0,-1,1,-1,3,-1,0,0,1,5,-2,0,1,-1,3,-2,1,0,-1,

%U 1,-1,1,-1,1,1,0,0,0,0,-1,4,0,0,0,-1,1,9,0,0

%N Irregular triangle read by rows: row n gives coefficients of basis polynomial {n,k} expressed in terms of binomial coefficients, high order terms first.

%H Peter J. C. Moses, <a href="/A263848/b263848.txt">First 300 rows.</a>

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/0801.0072">On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure</a>, arXiv|math.CO/0801.0072, 2007-2010. See Appendix.

%H Vladimir Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/m1/m1.Abstract.html">The number of permutations with prescribed up-down structure as a function of two variables</a>, INTEGERS, 12 (2012), #A1.

%H V. Shevelev and J. Spilker, <a href="http://dx.doi.org/10.4171/EM/229">Up-down coefficients for permutations</a>, Elemente der Mathematik, Vol. 68 (2013), no. 3, 115-127.

%e Triangle begins:

%e   1,

%e   1, -1,

%e   1,  0, -1,

%e   1, -1,  1,

%e   1,  0,  0, -1,

%e   2,  0, -1,  1,

%e   2, -1,  0,  1,

%e   1, -1,  1, -1,

%e   1,  0,  0,  0, -1,

%e   3,  0,  0, -1,  1,

%e   ...

%Y Cf. A133457, A264613.

%K sign,tabf,more

%O 0,14

%A _N. J. A. Sloane_, Nov 15 2015

%E More terms from _Peter J. C. Moses_, Dec 12 2015