

A035101


E.g.f. x*(c(x/2)1)/(12*x), where c(x) = g.f. for Catalan numbers A000108.


6



0, 1, 9, 87, 975, 12645, 187425, 3133935, 58437855, 1203216525, 27125492625, 664761133575, 17600023616175, 500706514833525, 15234653491682625, 493699195087473375, 16977671416936605375, 617528830880480644125, 23687738668934964248625
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OFFSET

1,3


COMMENTS

2nd column of triangular array A035342 whose first column is given by A001147(n), n >= 1. Recursion: a(n) = 2*n*a(n1)+ A001147(n1), n >= 2, a(1)=0.
a(n) gives the number of organically labeled forests (sets) with two rooted ordered trees with n nonroot 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, nonroot vertices) is increasing.  Wolfdieter Lang, Aug 07 2007
a(n), n>=2, enumerates unordered nvertex forests composed of two plane (ordered) ternary (3ary) 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.
a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly 1 of the remaining n1 chords are contained within the marked chord, see [Young].  Donovan Young, Aug 11 2020


LINKS

Robert Israel, Table of n, a(n) for n = 1..370
Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of LowDegreeTruncated Rational Multifunctions. Ib. Integrals of Materncorrelation functions for all oddhalfinteger class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.


FORMULA

a(n) = n!*A008549(n1)/2^(n1) = n!(4^(n1)binomial(2*n, n)/2)/2^(n1).
a(n) = (2n2)*a(n1) + A129890(n2).  Philippe Deléham, Oct 28 2013
a(n) = n!*2^(n1)  A001147(n) = A002866(n)  A001147(n).  Peter Bala, Sep 11 2015
a(n) = 2*(n1)*(2*n3)*a(n2)+(4*n3)*a(n1).  Robert Israel, Sep 11 2015


EXAMPLE

a(2)=1 for the forest: {r11, r22} (with root labels r1 and r2). The order between the components of the forest is irrelevant (like for sets).
a(3)=9 increasing ternary 2forest with n=3 vertices: there are three 2forests (the one vertex tree together with any of the three different 2vertex trees) each with three increasing labelings.  Wolfdieter Lang, Sep 14 2007


MAPLE

F:= gfun:rectoproc({(4*n^2+6*n+2)*a(n)+(4*n5)*a(n+1)+a(n+2), a(1)=0, a(2)=1, a(3)=9}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Sep 11 2015


MATHEMATICA

Table[Round [n! (4^(n  1)  Binomial[2 n, n]/2)/2^(n  1)], {n, 1, 20}] (* Vincenzo Librandi, Sep 12 2015 *)


PROG

(MAGMA) I:=[0, 1, 9]; [n le 3 select I[n] else  2*(n1)*(2*n3)*Self(n2)+(4*n3)*Self(n1): n in [1..30]]; // Vincenzo Librandi, Sep 12 2015
(PARI) a(n) = n!*(4^(n1)binomial(2*n, n)/2)/2^(n1);
vector(40, n, a(n)) \\ Altug Alkan, Oct 01 2015


CROSSREFS

Cf. A000108, A002866, A008549, A336599.
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.
Sequence in context: A223277 A267265 A152264 * A245491 A160466 A015583
Adjacent sequences: A035098 A035099 A035100 * A035102 A035103 A035104


KEYWORD

easy,nonn


AUTHOR

Wolfdieter Lang


STATUS

approved



