login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A286432 Numbers of labeled rooted Greg trees (A005264) with n nodes and root degree 2. 1
0, 1, 12, 151, 2545, 54466, 1417318, 43472780, 1536228588, 61466251616, 2746907348768, 135619260805568, 7331022129923648, 430638151053316480, 27315015477709844352, 1860627613021322933248, 135465573609158928964096, 10498038569346091127451136, 862792664850194915870874112 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Numbers of rooted Greg trees with 2 subtrees below root given m labeled nodes (lead index). Among all trees at the same index (see sequence A005264) root bifurcating trees play a central role in philological discourse on the reconstruction of manuscript genealogies. Labeled nodes represent surviving manuscripts, unlabeled nodes hypothetical ones. See also stemmatology/stemmatics, Bédier's paradox.

REFERENCES

J. Bédier. La tradition manuscrite du Lai de l'Ombre: Réflexions sur l'Art d' Éditer les Anciens Textes. Romania 394 (1928), 161-196/321-356.

C. Flight. How many stemmata? Manuscripta 34(2), 1990, 122-128.

W. Hering. Zweispaltige Stemmata. Philologus-Zeitschrift für antike Literatur und ihre Rezeption 111(1-2), (1967), 170-185.

P. Maas. Textkritik. 4. Auflage. Leipzig: Teubner. 1960.

LINKS

Armin Hoenen, Table of n, a(n) for n = 1..245

Armin Hoenen, S. Eger and R. Gehrke, How many stemmata with root degree k?, Proceedings of MOL 2017, 2017.

FORMULA

T_{m,2} = Sum_{n >= 0} T_{m,n,2}, where T_{m,n,k} = (m/k!) * Sum_{(s,p) in C((m-1,n),k)} (binomial(m-1,s) F(s,p)) + (1/k!) * Sum_{(s,p) in C((m,n-1),k)} (binomial(m,s) F(s,p)), with F(s,p) = Product_{1..k} (g(s_i,p_i)), here g(m,n) = numbers of rooted Greg trees, see (A005264) with m labeled and n unlabeled nodes. s and p are tuples with k elements where each s_i >= 1 and for each p_i : 0 <= p_i < s_i; first term in T_{m,n,k} gives the number of trees with a labeled root, second those for root unlabeled.

EXAMPLE

For n=3, T_{3,2} is T_{3,0,2} + T_{3,1,2} + T_{3,2,2} where T_{3,0,2} = (3/2) * (binomial(2,(1,1)) * product(g(1,0)*g(1,0))) + 0 = 3; T_{3,1,2} = 0 + 1/2 * ((binomial(3,(2,1)) * product(g(2,0)*g(1,0))) + (binomial(3,(1,2)) * product(g(1,0)*g(2,0)))) = 6 and T_{3,2,2} = 0 + (1/2) * ((binomial(3,(2,1)) * product(g(2,1)*g(1,0))) + (binomial(3,(1,2)) * product(g(1,0)*g(2,1)))) = 3; 3 + 6 + 3 =12.

CROSSREFS

Cf. A005264, number of labeled rooted Greg trees with n nodes.

Cf. A005263, unrooted Greg trees, according to Flight (1990) can also serve as basis for computation of A005624.

Sequence in context: A293153 A015611 A127650 * A189548 A103759 A351526

Adjacent sequences: A286429 A286430 A286431 * A286433 A286434 A286435

KEYWORD

nonn

AUTHOR

Armin Hoenen, May 09 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 26 14:22 EST 2022. Contains 358362 sequences. (Running on oeis4.)