A236396 -id:A236396 - cp's OEIS frontend

A034855 Triangle read by rows giving number of rooted labeled trees with n >= 2 nodes and height d >= 1.

2, 3, 6, 4, 36, 24, 5, 200, 300, 120, 6, 1170, 3360, 2520, 720, 7, 7392, 38850, 43680, 22680, 5040, 8, 50568, 475776, 757680, 551040, 221760, 40320, 9, 372528, 6231960, 13747104, 12836880, 7136640, 2358720, 362880, 10, 2936070, 87530400, 264181680

Offset: 2

N. J. A. Sloane

			2;
3,    6;
4,   36,    24;
5,  200,   300,   120;
6, 1170,  3360,  2520,   720;
7, 7392, 38850, 43680, 22680, 5040;

Alois P. Heinz, Rows n = 2..101, flattened
Marko Riedel, Counting the number of rooted trees of a certain height
Marko Riedel, Maple code for sequence (EGF)
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478. [broken link]
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A001854, A234953, A000435, A236396.

gf:= proc(k) gf(k):= `if`(k=0, x, x*exp(gf(k-1))) end:
A:= proc(n, k) A(n, k):= n!*coeff(series(gf(k), x, n+1), x, n) end:
T:= (n, d)-> A(n, d) -A(n, d-1):
seq(seq(T(n, d), d=1..n-1), n=2..12);  # Alois P. Heinz, Sep 21 2012

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k - 1]]; a[n_, k_] := n!*Coefficient[ Series[gf[k], {x, 0, n + 1}], x, n]; t[n_, d_] := a[n, d] - a[n, d - 1]; Table[t[n, d], {n, 2, 12}, {d, 1, n - 1}] // Flatten (* Jean-François Alcover, Jan 15 2013, translated from Alois P. Heinz's Maple program *)

Riordan reference gives recurrence.

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004

A001854 Total height of all rooted trees on n labeled nodes.

Original entry on oeis.org

0, 2, 15, 148, 1785, 26106, 449701, 8927192, 200847681, 5053782070, 140679853941, 4293235236324, 142553671807729, 5116962926162738, 197459475792232725, 8152354312656732976, 358585728464893234305, 16741214317684425260142, 826842457727306803110997, 43073414675338753123113980

Offset: 1

N. J. A. Sloane

nonn

Take any one of the n^(n-1) rooted trees on n labeled nodes, compute its height (maximal edge distance to root), sum over all trees.

Theorem [Renyi-Szekeres, (4,7)]. The average height if the tree is chosen at random is sqrt(2*n*Pi). - David desJardins, Jan 20 2017

Rényi, A., and G. Szekeres. "On the height of trees." Journal of the Australian Mathematical Society 7.04 (1967): 497-507. See (4.7).
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).

Alois P. Heinz, Table of n, a(n) for n = 1..387
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A000435, A034855, A236396.

Also A234953(n) = a(n)/n.

nn=20;a=NestList[ x Exp[#]&,x,nn];f[list_]:=Sum[list[[i]]*i,{i,1,Length[list]}];Drop[Map[f,Transpose[Table[Range[0,nn]!CoefficientList[Series[a[[i+1]]-a[[i]],{x,0,nn}],x],{i,1,nn-1}]]],1]  (* Geoffrey Critzer, Mar 14 2013 *)

a(n) = Sum_{k=1..n-1} A034855(n,k)*k. - Geoffrey Critzer, Mar 14 2013

A000435(n)/a(n) ~ 1/2 (see A000435 and the Renyi-Szekeres result mentioned in the Comments). - David desJardins, Jan 20 2017

More terms from Geoffrey Critzer, Mar 14 2013

A234953 Normalized total height of all rooted trees on n labeled nodes.

Original entry on oeis.org

0, 1, 5, 37, 357, 4351, 64243, 1115899, 22316409, 505378207, 12789077631, 357769603027, 10965667062133, 365497351868767, 13163965052815515, 509522144541045811, 21093278144993719665, 930067462093579181119, 43518024090910884374263, 2153670733766937656155699

Offset: 1

N. J. A. Sloane, Jan 14 2014

nonn

Equals A001854(n)/n. That is, similar to A001854, except here the root always has the fixed label 1.

This was in one of my thesis notebooks from 1964 (see the scans in A000435), but because it wasn't of central importance it was never added to the OEIS.

Alois P. Heinz, Table of n, a(n) for n = 1..387

Cf. A001854, A034855, A235595, A236396.

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k-1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; a[n_] := Sum[k*(a[n, k] - a[n, k-1]), {k, 1, n-1}]/n; Array[a, 20] (* Jean-François Alcover, Mar 18 2014, after Alois P. Heinz *)

from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*b(j - 1, h - 1)*b(n - j, h) for j in range(1, n + 1)])
def T(n, k): return b(n - 1, k - 1) - b(n - 1, k - 2)
def a(n): return sum([k*T(n, k) for k in range(1, n)])
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Aug 26 2017

a(n) = Sum_{k=1..n-1} k*A034855(n,k)/n = Sum_{k=1..n-1} k*A235595(n,k).

A235595 Triangle read by rows: the triangle in A034855, with the n-th row normalized by dividing it by n.

Original entry on oeis.org

1, 1, 2, 1, 9, 6, 1, 40, 60, 24, 1, 195, 560, 420, 120, 1, 1056, 5550, 6240, 3240, 720, 1, 6321, 59472, 94710, 68880, 27720, 5040, 1, 41392, 692440, 1527456, 1426320, 792960, 262080, 40320, 1, 293607, 8753040, 26418168, 30560544, 21213360, 9676800, 2721600, 362880, 1, 2237920, 119723130, 490458240, 691331760, 570810240, 323114400, 125798400, 30844800, 3628800

Offset: 2

N. J. A. Sloane, Jan 14 2014

T(n,k) is the number of forests of labeled rooted trees with n nodes and height k Cf. A210725. Equivalently, T(n,k) is the number of nilpotent partial functions on [n] with index k+1. - Geoffrey Critzer, Nov 26 2021

			Triangle begins:
1.
1, 2,
1, 9, 6,
1, 40, 60, 24,
1, 195, 560, 420, 120,
1, 1056, 5550, 6240, 3240, 720,
1, 6321, 59472, 94710, 68880, 27720, 5040,
1, 41392, 692440, 1527456,1426320, 792960, 262080, 40320,
1, 293607, 8753040, 26418168, 30560544, 21213360, 9676800, 2721600, 362880,
...

Alois P. Heinz, Rows n = 2..142, flattened

Cf. A000435, A034855, A001854, A210725, A234953, A235596, A236396.

Row sums give A000272.

b:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
      binomial(n-1, j-1)*j*b(j-1, h-1)*b(n-j, h), j=1..n))
    end:
T:= (n,k)-> b(n-1, k-1)-b(n-1, k-2):
seq(seq(T(n, d), d=1..n-1), n=2..12);  # Alois P. Heinz, Aug 21 2017

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k-1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; t[n_, k_] := (a[n, k] - a[n, k-1])/n; Table[t[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Mar 18 2014, after Alois P. Heinz *)

from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*b(j - 1, h - 1)*b(n - j, h) for j in range(1, n + 1)])
def T(n, k): return b(n - 1, k - 1) - b(n - 1, k - 2)
for n in range(2, 13): print([T(n, d) for d in  range(1, n)]) # Indranil Ghosh, Aug 26 2017, after Maple code

A234953(n) = Sum_{k=1..n} k*T(n,k).

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A034855 Triangle read by rows giving number of rooted labeled trees with n >= 2 nodes and height d >= 1.

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A001854 Total height of all rooted trees on n labeled nodes.

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A234953 Normalized total height of all rooted trees on n labeled nodes.

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A235595 Triangle read by rows: the triangle in A034855, with the n-th row normalized by dividing it by n.

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Maple

Mathematica

Python

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