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
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