A007827 -id:A007827 - cp's OEIS frontend

A271205 Number T(m,n) of series-reduced free trees with n nodes of which exactly m >= 3 are leaves, m+1 <= n <= 2m-2.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 4, 4, 2, 1, 3, 6, 10, 8, 4, 1, 3, 9, 17, 22, 15, 6, 1, 4, 12, 30, 47, 53, 32, 11, 1, 4, 16, 44, 91, 127, 121, 66, 18, 1, 5, 20, 67, 158, 282, 346, 292, 142, 37, 1, 5, 25, 91, 258, 539, 841, 921, 688, 306, 66, 1, 6, 30, 126, 397, 978, 1804, 2498, 2456, 1662, 672, 135, 1, 6, 36, 163, 588, 1636, 3550, 5856, 7260, 6489, 3978, 1483, 265, 1, 7, 42, 213, 838, 2638, 6495, 12554, 18636, 20946, 17082, 9629, 3316, 552, 1, 8

Offset: 3

Stephan Beyer, Apr 01 2016

The sequence of row sums a(m) = Sum_{n} T(m,n) is A007827.

The sequence of column sums a(n) = Sum_{m} T(m,n) is A000014.

			m\n | 3 4 5 6 7 8 9 10 11 12 13 14 15 16  17  18 19 20
-------------------------------------------------------
3   | . 1 . . . . .  .  .  .  .  .  .  .   .   .  .  .
4   | . . 1 1 . . .  .  .  .  .  .  .  .   .   .  .  .
5   | . . . 1 1 1 .  .  .  .  .  .  .  .   .   .  .  .
6   | . . . . 1 2 2  2  .  .  .  .  .  .   .   .  .  .
7   | . . . . . 1 2  4  4  2  .  .  .  .   .   .  .  .
8   | . . . . . . 1  3  6 10  8  4  .  .   .   .  .  .
9   | . . . . . . .  1  3  9 17 22 15  6   .   .  .  .
10  | . . . . . . .  .  1  4 12 30 47 53  32  11  .  .
11  | . . . . . . .  .  .  1  4 16 44 91 127 121 66 18

Stephan Beyer, Python code to generate the sequence on GitHub Gist

Cf. A000014, A007827.

Transpose of A271362.

A339786 Number of homeomorphically irreducible leaf colored trees with n leaves using exactly 3 colors.

Original entry on oeis.org

0, 0, 1, 9, 63, 412, 2673, 17571, 117365, 798819, 5530122, 38908380, 277750749, 2009160864, 14707923021, 108835512411, 813241695330, 6130521151377, 46584949832013, 356571373433217, 2747371943624943, 21296479544449677, 165994877608025730, 1300408539157086640

Offset: 1

Andrew Howroyd, Dec 18 2020

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=3 of A339780.

Cf. A007827, A339782, A339783, A339785.

my(N=25); (U(N,3) - 3*U(N,2) + 3*U(N,1))[2..1+N] \\ See A339780 for U(n, k).

a(n) = A339783(n) - 3*A339782(n) + 3*A007827(n).

A174224 Number of homeomorphism classes of stable curves in the moduli space of stable curves of genus g.

Original entry on oeis.org

0, 0, 7, 42, 379, 4555, 69808, 1281678, 27297406

Offset: 0

Stefano Maggiolo, Nov 27 2010

a(g) is also the number of stable graphs of type (g,0) (see reference).

			a(0) = a(1) = 0 because there are no stable curves of genus 0 or 1.

Melody T. Chan, Tropical curves and metric graphs, Ph. D. Dissertation, Univ. Calif. Berkeley, Spring 2012.
Melody Chan, Combinatorics of the tropical Torelli map, Algebra Number Theory, 6 (2012), 1133-1169.
Melody Chan, Graph complexes and the top-weight Euler characteristic of M_(g,n), Lecture Notes, Brown University (2020).
Pierre Deligne and David Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. No. 36, 1969, 75-109.
Stefano Maggiolo and Nicola Pagani, Boundary graph computer for the moduli space of stable pointed curves (includes program to compute a(n)).

Cf. A007827.

Row sums of triangle A366039.

A173383 Partial sums of A000014.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 6, 8, 12, 17, 27, 41, 67, 109, 187, 319, 568, 1013, 1855, 3416, 6404, 12075, 23056, 44265, 85737, 166918, 327094, 643843, 1273776, 2529846, 5045015, 10094831, 20267469, 40811048, 82413473, 166854359, 338648851, 688887026, 1404384063, 2868791176, 5871429462, 12038444069

Offset: 0

Jonathan Vos Post, Feb 17 2010

Number of series-reduced trees with <= n nodes.

Cf. A000014, A000055 (trees), A001678 (series-reduced planted trees), A007827 (series-reduced trees by leaves).

a(n) = Sum_{k=0..n} A000014(k).

a(3)=2 inserted and edited by Georg Fischer, Sep 14 2021

cp's OEIS Frontend

A271205 Number T(m,n) of series-reduced free trees with n nodes of which exactly m >= 3 are leaves, m+1 <= n <= 2m-2.

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A339786 Number of homeomorphically irreducible leaf colored trees with n leaves using exactly 3 colors.

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A174224 Number of homeomorphism classes of stable curves in the moduli space of stable curves of genus g.

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A173383 Partial sums of A000014.

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