

A264613


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


3



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
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OFFSET

1,1


COMMENTS

From Peter J. C. Moses, Dec 12 2015: (Start)
This appears to split into 3 sequences:
b(n) = 3*4^(n1)1, n>=1: 2,11,47,191,767,3071,12287,49151,...,
c(n) = 3*2^(2*n1)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

Table of n, a(n) for n=1..33.
Vladimir Shevelev, On the Basis Polynomials in the Theory of Permutations with Prescribed UpDown Structure, arXiv:0801.0072 [math.CO], 20072010. See Appendix.
Vladimir Shevelev, The number of permutations with prescribed updown structure as a function of two variables, INTEGERS, 12 (2012), #A1. (See Section 11, Problem 3.)
V. Shevelev and J. Spilker, Updown coefficients for permutations, Elemente der Mathematik, Vol. 68 (2013), no.3, 115127.


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


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]] (* JeanFrançois Alcover, Sep 06 2018 *)


CROSSREFS

Cf. A133457 (positive integer roots of {m,n}), A263848.
Sequence in context: A107679 A018846 A261578 * A285293 A246442 A056661
Adjacent sequences: A264610 A264611 A264612 * A264614 A264615 A264616


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Nov 28 2015


EXTENSIONS

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


STATUS

approved



