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A113882 Number of well-nested drawings of a rooted tree. 1
1, 1, 2, 9, 64, 605, 6996, 94556, 1452928, 24921765, 471091360, 9720039120, 217285778700, 5230874655578, 134929133296972, 3713182459524270, 108605754921052880, 3364866315332574493, 110099293819641466488, 3794219154973411079432, 137375263325254329836460 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The value a(n) also gives the number of non-crossing partitions on [n] such that (a) each block of the partition is a non-crossing partition itself (recursively) and (b) every partition in this recursion contains at least one singleton block (see A000108). Omitting the factor n in the equation for a(n) gives A110447.

REFERENCES

Manuel Bodirsky, Marco Kuhlmann and Mathias Mohl: Well-Nested Drawings as Models of Syntactic Structure, 10th Conference on Formal Grammar and 9th Meeting on Mathematics of Language, Edinburgh, Scotland, UK

LINKS

Table of n, a(n) for n=0..20.

Manuel Bodirsky, Marco Kuhlmann and Mathias Mohl: Well-Nested Drawings as Models of Syntactic Structure, 10th Conference on Formal Grammar and 9th Meeting on Mathematics of Language, Edinburgh, Scotland, UK

FORMULA

a(0) = a(1) = 1; a(n) = n * F(n-1), where F(0) = F(1) = 1,

F(n) = sum_{i=1}^{n} a(i) * F(n-i, i), where F(0, k) = 1; F(n, 1) = F(n),

F(n, k) = sum_{i=0}^{n} F(i) * F(n-i, k-1).

Contribution from Paul D. Hanna, Aug 08 2007 (revised Apr 28 2012): (Start)

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

a(n) = [x^(n-1)] A(x)^n for n>=1;

a(n) = (n+1)*A132070(n+1) for n>=0;

A(x) = x / Series_Reversion(x*G(x)) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A132070. (End)

EXAMPLE

a(5)=605 because there are 605 possibilities to form 5 nodes into a rooted tree and order the nodes of this tree such that no two subtrees interleave. Two subtrees t1, t2 interleave if their roots are (tree-)disjoint and there are four nodes l1, r1 from t1 and l2, r2 from t2 such that l1 < l2 < r1 < r2.

Comment from Paul D. Hanna, Aug 08 2007 (revised Apr 28 2012): (Start)

Illustrate a(n) = [x^(n-1)] A(x)^n by the following generating method.

Form a table of coefficients in powers of the g.f. A(x):

A(x)^1: [(1), 1, 2, 9, 64, 605, 6996, 94556, ...];

A(x)^2: [1, (2), 5, 22, 150, 1374, 15539, 206676, ...];

A(x)^3: [1, 3, (9), 40, 264, 2346, 25937, 339294, ...];

A(x)^4: [1, 4, 14, (64), 413, 3568, 38558, 495848, ...];

A(x)^5: [1, 5, 20, 95, (605), 5096, 53840, 680365, ...];

A(x)^6: [1, 6, 27, 134, 849, (6996), 72302, 897558, ...];

A(x)^7: [1, 7, 35, 182, 1155, 9345, (94556), 1152936, ...]; ...

then the coefficients along the main diagonal form the initial terms of this sequence. (End)

PROG

(PARI) {a(n)=local(G=1+x+2*x^2); for(k=0, n, G=1+x*deriv(serreverse(x/(G+x^2*O(x^n) )))); polcoeff(G, n)} - Paul D. Hanna, Aug 08 2007

CROSSREFS

Cf. A000108, A110447, A132070.

Sequence in context: A185897 A067297 A274394 * A059281 A269612 A269577

Adjacent sequences:  A113879 A113880 A113881 * A113883 A113884 A113885

KEYWORD

easy,nonn

AUTHOR

Marco Kuhlmann (kuhlmann(AT)ps.uni-sb.de), Jan 27 2006

EXTENSIONS

Initial term added and offset changed to 0 by Paul D. Hanna, Apr 28 2012.

STATUS

approved

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Last modified June 15 22:19 EDT 2019. Contains 324145 sequences. (Running on oeis4.)