OFFSET
0,3
COMMENTS
Apart from initial terms, same as A002378.
See A002378 for many more comments and references.
Denominators for the sequence with o.g.f. (1-x)*log(1-x). Numerators are given by 1 - 0^n - 2(C(1,n) - C(0,n)). Also denominators for the sequence with o.g.f. (1+x)*log(1+x). This sequence has numerators (-1)^n - 0^n + 2(C(1,n) - C(0,n)).
Also the denominator of the least distance between two adjacent Farey fractions of order n. The numerator is 1. - Robert G. Wilson v, Apr 13 2014
For n>0, a(n) are the Engel expansion of A096789. - Benedict W. J. Irwin, Dec 15 2016
Number of permutations of length n>=0 avoiding the partially ordered pattern (POP) {1>2} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second one. - Sergey Kitaev, Dec 08 2020
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..10000
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).
FORMULA
G.f.: (1-2*x+2*x^2+2*x^3-x^4) / (1-x)^3;
a(n) = 0^n + C(1, n) - C(0, n) + 2*C(n, 2).
MATHEMATICA
CoefficientList[Series[(1-2x+2x^2+2x^3-x^4)/(1-x)^3, {x, 0, 50}], x] (* or *) Denominator/@CoefficientList[Normal[Series[(1-x)Log[1-x], {x, 0, 50}]], x] (* Harvey P. Dale, Apr 20 2011 *)
PROG
(Magma) [0^n+Binomial(1, n)-Binomial(0, n)+2*Binomial(n, 2): n in [0..60]]; // Vincenzo Librandi, Dec 18 2016
CROSSREFS
KEYWORD
easy,nonn,frac
AUTHOR
Paul Barry, Feb 09 2005
STATUS
approved