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 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 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-François 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

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Last modified April 19 03:45 EDT 2021. Contains 343105 sequences. (Running on oeis4.)