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A035101 E.g.f. x*(c(x/2)-1)/(1-2*x), where c(x) = g.f. for Catalan numbers A000108.

%I A035101 #43 Sep 08 2022 08:44:52
%S A035101 0,1,9,87,975,12645,187425,3133935,58437855,1203216525,27125492625,
%T A035101 664761133575,17600023616175,500706514833525,15234653491682625,
%U A035101 493699195087473375,16977671416936605375,617528830880480644125,23687738668934964248625
%N A035101 E.g.f. x*(c(x/2)-1)/(1-2*x), where c(x) = g.f. for Catalan numbers A000108.
%C A035101 2nd column of triangular array A035342 whose first column is given by A001147(n), n >= 1. Recursion: a(n) = 2*n*a(n-1)+ A001147(n-1), n >= 2, a(1)=0.
%C A035101 a(n) gives the number of organically labeled forests (sets) with two rooted ordered trees with n non-root vertices. See the example a(3)=9 given in A035342. Organic labeling means that the vertex labels along the (unique) path from the root to any of the leaves (degree 1, non-root vertices) is increasing. - _Wolfdieter Lang_, Aug 07 2007
%C A035101 a(n), n>=2, enumerates unordered n-vertex forests composed of two plane (ordered) ternary (3-ary) trees with increasing vertex labeling. See A001147 (number of increasing ternary trees) and a D. Callan comment there. For a picture of some ternary trees see a W. Lang link under A001764.
%C A035101 a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly 1 of the remaining n-1 chords are contained within the marked chord, see [Young]. - _Donovan Young_, Aug 11 2020
%H A035101 Robert Israel, <a href="/A035101/b035101.txt">Table of n, a(n) for n = 1..370</a>
%H A035101 Selden Crary, Richard Diehl Martinez, Michael Saunders, <a href="https://arxiv.org/abs/1707.00705">The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters</a>, arXiv:1707.00705 [stat.ME], 2017, Table 2.
%H A035101 Alexander Kreinin, <a href="https://www.researchgate.net/profile/Alexander_Kreinin/publication/294260037">Integer Sequences and Laplace Continued Fraction</a>, Preprint 2016.
%H A035101 Alexander Kreinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Kreinin/kreinin4.html">Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity</a>, Journal of Integer Sequences, 19 (2016), #16.6.2.
%H A035101 Donovan Young, <a href="https://arxiv.org/abs/2007.13868">A critical quartet for queuing couples</a>, arXiv:2007.13868 [math.CO], 2020.
%F A035101 a(n) = n!*A008549(n-1)/2^(n-1) = n!(4^(n-1)-binomial(2*n, n)/2)/2^(n-1).
%F A035101 a(n) = (2n-2)*a(n-1) + A129890(n-2). - _Philippe Deléham_, Oct 28 2013
%F A035101 a(n) = n!*2^(n-1) - A001147(n) = A002866(n) - A001147(n). - _Peter Bala_, Sep 11 2015
%F A035101 a(n) = -2*(n-1)*(2*n-3)*a(n-2)+(4*n-3)*a(n-1). - _Robert Israel_, Sep 11 2015
%e A035101 a(2)=1 for the forest: {r1-1, r2-2} (with root labels r1 and r2). The order between the components of the forest is irrelevant (like for sets).
%e A035101 a(3)=9 increasing ternary 2-forest with n=3 vertices: there are three 2-forests (the one vertex tree together with any of the three different 2-vertex trees) each with three increasing labelings. - _Wolfdieter Lang_, Sep 14 2007
%p A035101 F:= gfun:-rectoproc({(4*n^2+6*n+2)*a(n)+(-4*n-5)*a(n+1)+a(n+2),a(1)=0,a(2)=1,a(3)=9},a(n),remember):
%p A035101 map(f, [$1..30]); # _Robert Israel_, Sep 11 2015
%t A035101 Table[Round [n! (4^(n - 1) - Binomial[2 n, n]/2)/2^(n - 1)], {n, 1, 20}] (* _Vincenzo Librandi_, Sep 12 2015 *)
%o A035101 (Magma) I:=[0,1,9]; [n le 3 select I[n] else - 2*(n-1)*(2*n-3)*Self(n-2)+(4*n-3)*Self(n-1): n in [1..30]]; // _Vincenzo Librandi_, Sep 12 2015
%o A035101 (PARI) a(n) = n!*(4^(n-1)-binomial(2*n, n)/2)/2^(n-1);
%o A035101 vector(40, n, a(n)) \\ _Altug Alkan_, Oct 01 2015
%Y A035101 Cf. A000108, A002866, A008549, A336599.
%Y A035101 Cf. A001147 (m=1 column of A035342). See a D. Callan comment there on the number of increasing ordered rooted trees on n+1 vertices.
%K A035101 easy,nonn
%O A035101 1,3
%A A035101 _Wolfdieter Lang_