%I #38 Jul 20 2023 15:17:38
%S 1,1,2,4,10,25,68,187,534,1544,4554,13576,40968,124681,382636,1182116,
%T 3674674,11483243,36057516,113701968,359927638,1143327888,3643379152,
%U 11643793399,37311200060,119852247220,385864664018,1244896820476
%N Revert transform of (1 + x - x^2) / (1 + x)^2.
%C a(n) is the number of ordered trees (A000108) with n edges in which every non-leaf non-root vertex has at most one leaf child. The g.f. A(x) is given by A(x)= x/(1-x B(x)) where B(x)=1+x+2x^2+4x^3+... is the g.f. for A143363. [_David Callan_, Aug 22 2014]
%C Conjecturally, the number of sequences (e(1), ..., e(n-1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) >= e(k). [Martinez and Savage, 2.10] - _Eric M. Schmidt_, Jul 17 2017
%H Wenqin Cao, Emma Yu Jin, and Zhicong Lin, <a href="https://doi.org/10.1016/j.dam.2019.01.035">Enumeration of inversion sequences avoiding triples of relations</a>, Discrete Applied Mathematics (2019); see also <a href="http://www.emmayujin.at/Pubs/CaoJinLin19.pdf">author's copy</a>
%H Megan A. Martinez and Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016.
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F Given g.f. A(x), then series reversion of B(x) = x + x * A(x) is -B(-x). - _Michael Somos_, Sep 07 2005
%F Given g.f. A(x), then B(x) = x + x * A(x) satisfies B(x) = x + C(x * B(x)) where C(x) is g.f. of A001764 offset 1.
%F D-finite with recurrence 5*n*(n-1)*(37*n-106)*a(n) -4*(n-1) *(74*n^2-323*n+288)*a(n-1) +16*(-74*n^3+508*n^2-1157*n+876)* a(n-2) +2*(2*n-5)*(37*n-69)*(n-4)*a(n-3)=0. - _R. J. Mathar_, Jun 24 2018
%F a(n) ~ (1+s)^2 / (2 * sqrt(Pi*(1+4*s)) * n^(3/2) * (s*(1 + s - s^2)/(1+s)^2)^(n - 1/2)), where s = 0.675130870566646070889621798150060480808032527677372732 = 2*cos(arctan(sqrt(37/27))/3)/sqrt(3) + 2*sin(arctan(sqrt(37/27))/3) - 1 is the root of the equation s^3 + 3*s^2 - s = 1. - _Vaclav Kotesovec_, Aug 17 2018
%e x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 25*x^6 + 68*x^7 + 187*x^8 + 534*x^9 + ...
%t a[1] = 1;
%t a[n_] := SeriesCoefficient[InverseSeries[x(1+x-x^2)/(1+x)^2 + x O[x]^n, x], {x, 0, n}];
%t Array[a, 28] (* _Jean-François Alcover_, Aug 17 2018, from PARI *)
%t CoefficientList[InverseSeries[Series[x*(1 + x - x^2)/(1 + x)^2, {x, 0, 30}], x], x] (* _Vaclav Kotesovec_, Aug 17 2018 *)
%o (PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( x * (1 + x - x^2) / (1 + x)^2 + x * O(x^n)), n))} /* _Michael Somos_, Jul 13 2003 */
%K nonn
%O 1,3
%A _Olivier Gérard_