A000524 -id:A000524 - cp's OEIS frontend

A008295 Triangle read by rows: T(n,k) is the number of partially labeled rooted trees with n vertices, k of which are labeled, 0 <= k <= n.

1, 1, 1, 1, 2, 2, 2, 5, 9, 9, 4, 13, 34, 64, 64, 9, 35, 119, 326, 625, 625, 20, 95, 401, 1433, 4016, 7776, 7776, 48, 262, 1316, 5799, 21256, 60387, 117649, 117649, 115, 727, 4247, 22224, 100407, 373895, 1071904, 2097152, 2097152

Offset: 0

N. J. A. Sloane

T(n, k) where n counts the vertices and 0 <= k <= n counts the labels. - Sean A. Irvine, Mar 22 2018

			Triangle begins with T(0,0):
n\k 0  1   2    3    4    5    6
0   1
1   1  1
2   1  2   2
3   2  5   9    9
4   4 13  34   64   64
5   9 35 119  326  625  625
6  20 95 401 1433 4016 7776 7776

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.

Sean A. Irvine, Table of n, a(n) for n = 0..527
J. Riordan, The numbers of labeled colored and chromatic trees, Acta Mathematica 97 (1957) 211.
Index entries for sequences related to rooted trees

Main diagonal is A000169.
Columns k=0..6 are A000081, A000107, A000524, A000444, A000525, A064781, A064782.
Cf. A034799.

m = 9; r[_] = 0;
Do[r[x_] = x Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, m}];
r[x_, y_] = -ProductLog[(-E^(-r[x])) r[x] - (r[x] y)/E^r[x]];
(CoefficientList[#, y] Range[0, Exponent[#, y]]!)& /@ CoefficientList[r[x, y] + O[x]^m, x] /. {} -> {1} // Flatten // Quiet (* Jean-François Alcover, Oct 23 2019 *)

E.g.f.: r(x,y) = T(n,k) * y^k * x^n / k! satisfies r(x,y) * exp(r(x)) = (1+y) * r(x) * exp(r(x,y)) where r(x) is the o.g.f. for A000081. - Sean A. Irvine, Mar 22 2018

More terms from Sean A. Irvine, Mar 22 2018

Name edited by Andrew Howroyd, Mar 23 2023

A000269 Number of trees with n nodes, 3 of which are labeled.

Original entry on oeis.org

3, 16, 67, 251, 888, 3023, 10038, 32722, 105228, 334836, 1056611, 3311784, 10322791, 32026810, 98974177, 304835956, 936147219, 2867586542, 8764280567, 26733395986, 81399821915, 247459136331, 751211286356, 2277496842016

Offset: 3

N. J. A. Sloane

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=3..200
Index entries for sequences related to trees

Column k=3 of A034799.

b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n+1 - j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[ b[k]*x^k, {k, 1, n}]; a[n_] := SeriesCoefficient[ B[n-1]^3 * (2*B[n-1]-3) / (B[n-1]-1)^3, {x, 0, n}]; Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Jan 27 2015 *)

G.f.: A(x) = B(x)^3*(3-2*B(x))/(1-B(x))^3, where B(x) is g.f. for rooted trees with n nodes, cf. A000081. - Vladeta Jovovic, Oct 19 2001

a(n) = A000524(n) - 2*A000243(n).

More terms, new description and formula from Christian G. Bower, Nov 15 1999

cp's OEIS Frontend

A008295 Triangle read by rows: T(n,k) is the number of partially labeled rooted trees with n vertices, k of which are labeled, 0 <= k <= n.

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