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A259845 a(0)=1, a(1)=3, and the INVERT transform of the sequence deletes the 3.

%I A259845 #26 Jun 12 2024 12:06:03
%S A259845 1,3,4,11,38,136,512,1993,7958,32420,134216,563030,2388092,10224320,
%T A259845 44127328,191783029,838623654,3686965308,16287624440,72262899994,
%U A259845 321852273332,1438540956048,6450223722816,29006443606746,130790584554748,591191800834696
%N A259845 a(0)=1, a(1)=3, and the INVERT transform of the sequence deletes the 3.
%C A259845 The sequence is N = 3 in an infinite set, with the first few being:
%C A259845 A086581, N = 0: (1, 0, 1, 2, 5, 13, 35, 97, ...)
%C A259845 A000108, N = 1: (1, 1, 2, 5, 14, 42, 132, ...)
%C A259845 A171199, N = 2: (1, 2, 3, 8, 25, 83, 289, ...)
%C A259845 ... The INVERT transforms of the sequences delete the second terms in the sequences.
%C A259845 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.
%H A259845 Alois P. Heinz, <a href="/A259845/b259845.txt">Table of n, a(n) for n = 0..1000</a>
%H A259845 Amya Luo, <a href="https://math.dartmouth.edu/theses/undergrad/2024/Luo-thesis.pdf">Pattern Avoidance in Nonnesting Permutations</a>, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
%F A259845 G.f.: A(x) = 1/(2*x) + x - sqrt(1 - 4*x - 4*x^2 + 4*x^4)/(2*x).
%e A259845 The INVERT transform of (1, 3, 4, 11, 38, 136, ...) is (1, 4, 11, 38, 136, ...).
%t A259845 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 *)
%Y A259845 Cf. A086581, A000108, A171199.
%K A259845 nonn,eigen
%O A259845 0,2
%A A259845 _Gary W. Adamson_, Jul 06 2015
%E A259845 More terms from _Alois P. Heinz_, Jul 07 2015