A227819 -id:A227819 - cp's OEIS frontend

A004111 Number of rooted identity trees with n nodes (rooted trees whose automorphism group is the identity group).

0, 1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 247, 548, 1226, 2770, 6299, 14426, 33209, 76851, 178618, 416848, 976296, 2294224, 5407384, 12780394, 30283120, 71924647, 171196956, 408310668, 975662480, 2335443077, 5599508648, 13446130438, 32334837886, 77863375126, 187737500013, 453203435319, 1095295264857, 2649957419351

Offset: 0

N. J. A. Sloane

The nodes are unlabeled.
There is a natural correspondence between rooted identity trees and finitary sets (sets whose transitive closure is finite); each node represents a set, with the children of that node representing the members of that set. When the set corresponding to an identity tree is written out using braces, there is one set of braces for each node of the tree; thus a(n) is also the number of sets that can be made using n pairs of braces. - Franklin T. Adams-Watters, Oct 25 2011
Shifts left under WEIGH transform. - Franklin T. Adams-Watters, Jan 17 2007
Is this the sequence mentioned in the middle of page 355 of Motzkin (1948)? - N. J. A. Sloane, Jul 04 2015. Answer from David Broadhurst, Apr 06 2022: The answer is No. Motzkin was considering a sequence asymptotic to Catalan(n)/(4*n), namely A006082, which begins 1, 1, 1, 2, 3, 6, 12, 27, ... but he miscalculated and got 1, 1, 1, 2, 3, 6, 12, 25, ... instead! - N. J. A. Sloane, Apr 06 2022

			The 2 identity trees with 4 nodes are:
     O    O
    / \   |
   O   O  O
       |  |
       O  O
          |
          O
These correspond to the sets {{},{{}}} and {{{{}}}}.
G.f.: x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 12*x^7 + 25*x^8 + 52*x^9 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 330.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 301 and 562.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 64, Eq. (3.3.15); p. 80, Problem 3.10.
D. E. Knuth, Fundamental Algorithms, 3rd Ed., 1997, pp. 386-388.
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 = 0..2500 (first 201 terms from T. D. Noe)
Joerg Arndt, All identity trees for n = 1..11.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
A. Genitrini, Full asymptotic expansion for Polya structures, arXiv:1605.00837 [math.CO], May 03 2016, p. 8.
Bernhard Gittenberger, Emma Yu Jin, Michael Wallner, On the shape of random Pólya structures, arXiv|1707.02144 [math.CO], 2017-2018; Discrete Math., 341 (2018), 896-911.
Frank Harary and Geert Prins, The number of homeomorphically irreducible trees and other species, Acta Math., 101 (1959), 141-162.
F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A, 20 (1975), 483-503.
F. Harary, R. W. Robinson and A. J. Schwenk, Corrigenda: Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A 41 (1986), p. 325.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 56.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
N. J. A. Sloane, Sketch showing trees with 2 through 6 nodes.
Index entries for sequences related to rooted trees

Cf. A000009, A000081, A000220, A196118, A196154, A196161, A227819.
Cf. A027750, A035056, A246169.
Column k=1 of A255517, A316074, A316101, A318757.

import Data.List (genericIndex)
a004111 = genericIndex a004111_list
a004111_list = 0 : 1 : f 1 [1] where
   f x zs = y : f (x + 1) (y : zs) where
            y = (sum $ zipWith (*) zs $ map g [1..]) `div` x
   g k = sum $ zipWith (*) (map (((-1) ^) . (+ 1)) $ reverse divs)
                           (zipWith (*) divs $ map a004111 divs)
                           where divs = a027750_row k
-- Reinhard Zumkeller, Apr 29 2014

A004111 := proc(n)
        spec := [ A, {A=Prod(Z,PowerSet(A))} ]:
        combstruct[count](spec, size=n) ;
end proc:
# second Maple program:
with(numtheory):
a:= proc(n) a(n):= `if`(n<2, n, add(a(n-k)*add(a(d)*d*
       (-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 15 2014

s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, -s[ n-k, k ] ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ] (* Robert A. Russell *)
a[ n_] := If[ n < 2, Boole[n == 1], Nest[ CoefficientList[ Normal[ Times @@ (Table[1 + x^k, {k, Length@#}]^#) + x O[x]^Length@#], x] &, {}, n - 1][[n]]]; (* Michael Somos, Jul 10 2014 *)
a[n_] := a[n] = Sum[a[n-k]*Sum[a[d]*d*(-1)^(k/d+1),{d, Divisors[k]}], {k, 1, n-1}]/(n-1); a[0]=0; a[1]=1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 02 2015 *)

N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1) * d * A[d]) * A[n-k+1] ) );
concat([0], A)
\\ Joerg Arndt, Jul 10 2014

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) d*a(d) ) * a(n-k+1). - Mitchell Harris, Dec 02 2004
G.f. satisfies A(x) = x*exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...). [Harary and Prins]
Also A(x) = Sum_{n >= 1} a(n)*x^n = x * Product_{n >= 1} (1+x^n)^a(n).
a(n) ~ c * d^n / n^(3/2), where d = A246169 = 2.51754035263200389079535..., c = 0.3625364233974198712298411097408713812865256408189512533230825639621448038... . - Vaclav Kotesovec, Aug 22 2014, updated Dec 26 2020

A034781 Triangle of number of rooted trees with n >= 2 nodes and height h >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 10, 18, 13, 5, 1, 1, 14, 38, 36, 19, 6, 1, 1, 21, 76, 93, 61, 26, 7, 1, 1, 29, 147, 225, 180, 94, 34, 8, 1, 1, 41, 277, 528, 498, 308, 136, 43, 9, 1, 1, 55, 509, 1198, 1323, 941, 487, 188, 53, 10, 1

Offset: 2

N. J. A. Sloane

			Triangle begins:
  1;
  1  1;
  1  2  1;
  1  4  3  1;
  1  6  8  4  1;
  1 10 18 13  5  1;
  1 14 38 36 19  6 1;
thus there are 10 trees with 7 nodes and height 2.

Alois P. Heinz, Rows n = 2..142, flattened
Marko Riedel, Counting the number of rooted trees of a certain height
Marko Riedel, Maple code for sequence (OGF)
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)
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Columns h=2-10 give: A000065, A000235, A000299, A000342, A000393, A000418, A000429, A126085, A245068.
T(2n,n) = A245102(n), T(2n+1,n) = A245103(n).
Row sums give A000081.
Cf. A001853, A227819, A375467.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
     add(binomial(b((i-1)$2, k-1)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
T:= (n, k)-> b((n-1)$2, k) -b((n-1)$2, k-1):
seq(seq(T(n, k), k=1..n-1), n=2..16);  # Alois P. Heinz, Jul 31 2013

Drop[Map[Select[#, # > 0 &] &,
   Transpose[
    Prepend[Table[
      f[n_] :=
       Nest[CoefficientList[
          Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x,
            0, 10}], x] &, {1}, n]; f[m] - f[m - 1], {m, 2, 10}],
Prepend[Table[1, {10}], 0]]]], 1] // Grid (* Geoffrey Critzer, Aug 01 2013 *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1 || k<1, 0, Sum[Binomial[b[i-1, i-1, k-1]+j-1, j]*b[n-i*j, i-1, k], {j, 0, n/i}]]]; T[n_, k_] := b[n-1, n-1, k]-b[n-1, n-1, k-1]; Table[T[n, k], {n, 2, 16}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 11 2014, after Alois P. Heinz *)

def A034781(n, k): return A375467(n, k) - A375467(n, k - 1)
for n in range(2, 10): print([A034781(n, k) for k in range(2, n + 1)])
# Peter Luschny, Aug 30 2024

Reference gives recurrence.

T(n, k) = A375467(n, k) - A375467(n, k - 1). - Peter Luschny, Aug 30 2024

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 19 2003

A038081 Number of rooted identity trees of height n. Sets of rank n.

Original entry on oeis.org

1, 1, 2, 12, 65520

Offset: 0

Christian G. Bower, Jan 04 1999

Next term is 2^65536 - 65536.

Eric Weisstein's World of Mathematics, Rank.
Index entries for sequences related to rooted trees

Cf. A004111, A038082-A038093, A048828.
Differences of A014221.
Column sums of A227819.

A038093 Number of nodes in largest rooted identity tree of height n.

Original entry on oeis.org

1, 2, 4, 11, 97, 3211265

Offset: 0

Christian G. Bower, Jan 04 1999

nonn

The next term is 19735 digits long, which is too large even for a b-file.

Also, the sequence gives the number of pairs of braces in the n-th iteration of the von Neumann universe. - Adam P. Goucher, Aug 18 2013

			For n = 3, the n-th iteration of the von Neumann universe is V3 = {{}, {{}}, {{{}}}, {{},{{}}}}, which has a(3) = 11 pairs of braces.

Adam P. Goucher, Article including the first five iterations of the von Neumann universe, "Complex Projective 4-Space" blog.
Index entries for sequences related to rooted trees

Cf. A038082, A038083, A038084, A038085, A038086, A038087, A038088, A038089, A038090, A038091, A038092, A229403, A229404.

Cf. A227819.

h:= (b, k)-> `if`(k=0, 1, b^h(b, k-1)):
a:= proc(n) option remember; `if`(n=0, 1,
       1+(1+a(n-1))/2*h(2, n-1))
    end:
seq(a(n), n=0..5);  # Alois P. Heinz, Aug 25 2017

Map[#[[1]]&,NestList[{(#[[1]]+1)*(2^#[[2]])/2+1,2^#[[2]]}&,{1,0},6]] (* Adam P. Goucher, Aug 18 2013 *)

Recurrence relation: a(n+1) = (a(n) + 1)*(2^^n)/2 + 1 where 2^^n is Knuth's up-arrow notation. - Adam P. Goucher, Aug 18 2013

a(6) from Adam P. Goucher, Aug 18 2013

A291529 Number F(n,h,t) of forests of t (unlabeled) rooted identity trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 5, 1, 0, 0, 5, 4, 0, 0, 4, 1, 0, 1, 0

Offset: 0

Alois P. Heinz, Aug 25 2017

Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A227819.

Positive column sums per layer give A227774.

			n h\t: 0 1 2 3 4 5 : A227819 : A227774   : A004111
-----+-------------+---------+-----------+--------
0 0  : 1           :         :           : 1
-----+-------------+---------+-----------+--------
1 0  : 0 1         :       1 : .         :
1 1  : 0           :         : 1         : 1
-----+-------------+---------+-----------+--------
2 0  : 0 0 0       :       0 : . .       :
2 1  : 0 1         :       1 : .         :
2 2  : 0           :         : 1 0       : 1
-----+-------------+---------+-----------+--------
3 0  : 0 0 0 0     :       0 : . . .     :
3 1  : 0 0 1       :       1 : . .       :
3 2  : 0 1         :       1 : .         :
3 3  : 0           :         : 1 1 0     : 2
-----+-------------+---------+-----------+--------
4 0  : 0 0 0 0 0   :       0 : . . . .   :
4 1  : 0 0 0 0     :       0 : . . .     :
4 2  : 0 1 1       :       2 : . .       :
4 3  : 0 1         :       1 : .         :
4 4  : 0           :         : 2 1 0 0   : 3
-----+-------------+---------+-----------+--------
5 0  : 0 0 0 0 0 0 :       0 : . . . . . :
5 1  : 0 0 0 0 0   :       0 : . . . .   :
5 2  : 0 0 2 0     :       2 : . . .     :
5 3  : 0 2 1       :       3 : . .       :
5 4  : 0 1         :       1 : .         :
5 5  : 0           :         : 3 3 0 0 0 : 6
-----+-------------+---------+-----------+--------

Alois P. Heinz, Layers n = 0..48, flattened

Cf. A000007, A004111, A227774, A227819, A291203, A291204, A291336, A291532, A291559.

b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0 or i=1,
      `if`(n<2, x^(t*n), 0), b(n, i-1, t, h)+add(x^(t*j)*binomial(
       b(i-1$2, 0, h-1), j)*b(n-i*j, i-1, t, h), j=1..n/i)))
    end:
g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..10);

b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0 || i == 1, If[n < 2, x^(t*n), 0], b[n, i - 1, t, h] + Sum[x^(t*j)*Binomial[b[i - 1, i - 1, 0, h - 1], j]*b[n - i*j, i - 1, t, h], {j, 1, n/i}]]];
g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h - 1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[F[n, h, t], {n, 0, 10}, {h, 0, n}, {t, 0, n - h}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)

Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A004111(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A291532(n).
Sum_{h=0..n-2} Sum_{t=1..n-1-h} (h+1) * F(n-1,h,t) = A291559(n).
F(n,0,0) = A000007(n).

A166830 Number of n X 3 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nonincreasing order.

Original entry on oeis.org

2, 8, 18, 33, 54, 82, 118, 163, 218, 284, 362, 453, 558, 678, 814, 967, 1138, 1328, 1538, 1769, 2022, 2298, 2598, 2923, 3274, 3652, 4058, 4493, 4958, 5454, 5982, 6543, 7138, 7768, 8434, 9137, 9878, 10658, 11478, 12339, 13242

Offset: 1

R. H. Hardin, Oct 21 2009

nonn

			All solutions for n=3
...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1
...1.1.1...1.1.1...1.1.1...2.1.1...2.1.1...2.1.1...2.2.1...2.2.1...2.2.2
...2.1.1...2.2.1...2.2.2...2.1.1...2.2.1...2.2.2...2.2.1...2.2.2...2.2.2
------
...2.1.1...2.1.1...2.1.1...2.1.1...2.1.1...2.1.1...2.2.1...2.2.1...2.2.1
...2.1.1...2.1.1...2.1.1...2.2.1...2.2.1...2.2.2...2.2.1...2.2.1...2.2.2
...2.1.1...2.2.1...2.2.2...2.2.1...2.2.2...2.2.2...2.2.1...2.2.2...2.2.2

lst={};Do[AppendTo[lst,n*(n+1)*(n+2)/6-2],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)

Empirical: a(n) = (n^3+6*n^2+11*n-6)/6.
a(n) = A167772(n+3,n). - Philippe Deléham, Nov 11 2009
a(n) = A227819(n+6,n+2). - Alois P. Heinz, Sep 22 2013
Empirical: a(n) = floor(A000292(n+1)^3/(A000292(n+1) + 1)^ 2). - Ivan N. Ianakiev, Nov 05 2013
From G. C. Greubel, May 25 2016: (Start)
Empirical G.f.: (-1 + 6*x - 6*x^2 + 2*x^3)/(1 - x)^4 + 1.
Empirical E.g.f.: (1/6)*(-6 + 18*x + 9*x^2 + x^3)*exp(x) + 1. (End)

A038088 Number of n-node rooted identity trees of height 4.

Original entry on oeis.org

1, 3, 5, 8, 12, 17, 23, 32, 41, 52, 66, 83, 102, 124, 152, 181, 216, 255, 299, 346, 400, 458, 521, 588, 659, 735, 814, 896, 979, 1067, 1151, 1239, 1324, 1407, 1486, 1564, 1635, 1700, 1759, 1809, 1853, 1887, 1912, 1925, 1932, 1925, 1912, 1887, 1853

Offset: 5

Christian G. Bower, Jan 04 1999

Alois P. Heinz, Table of n, a(n) for n = 5..97
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

Column k=4 of A227819.

weigh:= proc(p) proc(n) local x, k; coeff(series(mul((1+x^k)^p(k), k=1..n), x, n+1), x, n) end end: wsh:= p-> n-> weigh(p)(n-1): f:= n-> `if`(n>0 and n<12, [1$3, 2$5, 1$3][n], 0): a:= wsh(f)-f: seq(a(n), n=5..97); # Alois P. Heinz, Sep 10 2008

f[n_]:=Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,50}],x]&,{1},n];Drop[f[4]-PadRight[f[3],Length[f[4]]],4] (* Geoffrey Critzer, Aug 01 2013 *)

a(n) = A038083(n) - A038082(n).

A038089 Number of n-node rooted identity trees of height 5.

Original entry on oeis.org

1, 4, 9, 18, 34, 61, 108, 187, 318, 528, 871, 1417, 2288, 3662, 5825, 9203, 14471, 22639, 35266, 54725, 84607, 130379, 200287, 306787, 468607, 713970, 1085060, 1645181, 2488771, 3756778, 5658871, 8506900, 12763178, 19112874, 28568961, 42627442, 63493739

Offset: 6

Christian G. Bower, Jan 04 1999

The number of terms with a(n)>0 is A038093(5)-5 = 3211260. - Alois P. Heinz, Sep 22 2013

Alois P. Heinz, Table of n, a(n) for n = 6..1000
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

Column k=5 of A227819.

weigh:= proc(p) proc(n) local x, k; coeff(series(mul((1+x^k)^p(k), k=1..n), x, n+1), x, n) end end: wsh:= p-> n-> weigh(p)(n-1): f:= n-> `if`(n>0 and n<12, [1$3, 2$5, 1$3][n], 0): a:= (wsh@@2)(f)-wsh(f): seq(a(n), n=6..40);  # Alois P. Heinz, Sep 10 2008

f[n_]:=Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,50}],x]&,{1},n];Drop[f[5]-PadRight[f[4], Length[f[5]]],5] (* Geoffrey Critzer, Aug 01 2013 *)

A038084 - A038083.

A038090 Number of n-node rooted identity trees of height 6.

Original entry on oeis.org

1, 5, 14, 33, 72, 149, 301, 599, 1170, 2254, 4288, 8081, 15087, 27971, 51500, 94293, 171724, 311328, 562023, 1010819, 1811676, 3236959, 5766793, 10246734, 18162241, 32119542, 56682671, 99833464, 175509158, 308014335, 539675744, 944115593, 1649236884

Offset: 7

Christian G. Bower, Jan 04 1999

The number of terms with a(n)>0 is A038093(6) - 6. - Alois P. Heinz, Sep 22 2013

Alois P. Heinz, Table of n, a(n) for n = 7..1000
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

Column k=6 of A227819.

weigh:= proc(p) proc(n) local x, k; coeff(series(mul((1+x^k)^p(k), k=1..n), x,n+1), x,n) end end: wsh:= p-> n-> weigh(p)(n-1): f:= n-> `if`(n>0 and n<12, [1$3, 2$5, 1$3][n], 0): a:= (wsh@@3)(f)-(wsh@@2)(f): seq(a(n), n=7..37);  # Alois P. Heinz, Sep 10 2008

f[n_]:=Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,50}],x]&,{1},n];Drop[f[6]-PadRight[f[5],Length[f[6]]],6] (* Geoffrey Critzer, Aug 01 2013 *)

A038085 - A038084.

A038091 Number of n-node rooted identity trees of height 7.

Original entry on oeis.org

1, 6, 20, 54, 132, 303, 672, 1460, 3120, 6575, 13707, 28296, 57938, 117764, 237878, 477781, 954910, 1899930, 3765054, 7433724, 14628436, 28698388, 56143591, 109550807, 213251179, 414190801, 802808056, 1553046868, 2998986556, 5781366468, 11127506290

Offset: 8

Christian G. Bower, Jan 04 1999

The number of terms with a(n)>0 is A038093(7) - 7. - Alois P. Heinz, Sep 22 2013

Alois P. Heinz, Table of n, a(n) for n = 8..1000
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

Column k=7 of A227819.

weigh:= proc(p) proc(n) local x,k; coeff(series(mul((1+x^k)^p(k), k=1..n), x,n+1), x,n) end end: wsh:= p-> n-> weigh(p)(n-1): f:= n-> `if`(n>0 and n<12, [1$3,2$5,1$3][n], 0): a:= (wsh@@4)(f)-(wsh@@3)(f): seq(a(n), n=8..36);  # Alois P. Heinz, Sep 10 2008

f[n_]:=Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,50}],x]&,{1},n];Drop[f[7]-PadRight[f[6],Length[f[7]]],7] (* Geoffrey Critzer, Aug 01 2013 *)

A038086 - A038085.

cp's OEIS Frontend

A004111 Number of rooted identity trees with n nodes (rooted trees whose automorphism group is the identity group).

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A034781 Triangle of number of rooted trees with n >= 2 nodes and height h >= 1.

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A038081 Number of rooted identity trees of height n. Sets of rank n.

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A038093 Number of nodes in largest rooted identity tree of height n.

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A291529 Number F(n,h,t) of forests of t (unlabeled) rooted identity trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

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A166830 Number of n X 3 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nonincreasing order.

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A038088 Number of n-node rooted identity trees of height 4.

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A038089 Number of n-node rooted identity trees of height 5.