OFFSET
0,2
COMMENTS
The sequence is N = 3 in an infinite set, with the first few being:
A086581, N = 0: (1, 0, 1, 2, 5, 13, 35, 97, ...)
A000108, N = 1: (1, 1, 2, 5, 14, 42, 132, ...)
A171199, N = 2: (1, 2, 3, 8, 25, 83, 289, ...)
... The INVERT transforms of the sequences delete the second terms in the sequences.
The g.f. was contributed by Paul D. Hanna: From the definition of the INVERT transform, 1/(1 - x*A) = A - (N-1)*x. Thus, (1 + (N-1)*x - (1 + (N-1)*x^2)*A) + x*A^2 = 0. The g.f. follows, below.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
FORMULA
G.f.: A(x) = 1/(2*x) + x - sqrt(1 - 4*x - 4*x^2 + 4*x^4)/(2*x).
EXAMPLE
The INVERT transform of (1, 3, 4, 11, 38, 136, ...) is (1, 4, 11, 38, 136, ...).
MATHEMATICA
CoefficientList[Series[1/(2*x) + x - Sqrt[1 - 4*x - 4*x^2 + 4*x^4]/(2*x), {x, 0, 25}], x] (* Michael De Vlieger, Jun 12 2024 *)
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
Gary W. Adamson, Jul 06 2015
EXTENSIONS
More terms from Alois P. Heinz, Jul 07 2015
STATUS
approved