login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Table of n, a(n) for n=1..33.

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.

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]] (* Jean-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 14:47 EST 2019. Contains 329806 sequences. (Running on oeis4.)