A264613 Numbers n such that the Shevelev polynomial {m, n} has a root at m = -1.

2, 5, 8, 11, 23, 32, 47, 95, 128, 191, 383, 512, 767, 1535, 2048, 3071, 6143, 8192, 12287, 24575, 32768, 49151, 98303, 131072, 196607, 393215, 524288, 786431, 1572863, 2097152, 3145727, 6291455, 8388608

Offset: 1

Author

N. J. A. Sloane, Nov 28 2015

Keywords

• nonn

Comments

From Peter J. C. Moses, Dec 12 2015: (Start)

This appears to split into 3 sequences:

b(n) = 3*4^(n-1)-1, n>=1: 2,11,47,191,767,3071,12287,49151,...,

c(n) = 3*2^(2*n-1)-1, n>=1: 5,23,95,383,1535,6143,24575,98303,...,

d(n) = 2^(2*n+1), n>=1: 8,32,128,512,2048,8192,32768,...;

If this is true, then the next few terms of the sequence are 12582911, 25165823, 33554432, 50331647, 100663295, ...

(End)

Links

• Vladimir Shevelev, On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure, arXiv:0801.0072 [math.CO], 2007-2010. See Appendix.
• Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. (See Section 11, Problem 3.)
• V. Shevelev and J. Spilker, Up-down coefficients for permutations, Elemente der Mathematik, Vol. 68 (2013), no.3, 115-127.

Crossrefs

Cf. A133457 (positive integer roots of {m,n}), A263848.

Programs

Mathematica

upDown[n_, k_] := upDown[n, k] = Module[{t, m}, t = Flatten[ Reverse[ Position[ Reverse[ IntegerDigits[k, 2]], 1]]]; m = Length[t]; (-1)^m + Sum[upDown[t[[j]], k - 2^(t[[j]] - 1)]*Binomial[n, t[[j]]], {j, 1, m}]];
Reap[For[k = 2, k <= 2^15, k++, If[(upDown[n, k] /. n -> -1) == 0, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Sep 06 2018 *)

Formula

Conjectured g.f.: (2 + x*(5 + x*(8 + x*(1 + (-2 - 8*x)* x)))) / (1 + x^3*(-5 + 4*x^3)). - Peter J. C. Moses, Dec 12 2015

Extensions

More terms (starting at a(6)) from Peter J. C. Moses, Dec 12 2015