A255517 -id:A255517 - 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

A242249 Number A(n,k) of rooted trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 7, 4, 0, 0, 1, 4, 15, 26, 9, 0, 0, 1, 5, 26, 82, 107, 20, 0, 0, 1, 6, 40, 188, 495, 458, 48, 0, 0, 1, 7, 57, 360, 1499, 3144, 2058, 115, 0, 0, 1, 8, 77, 614, 3570, 12628, 20875, 9498, 286, 0, 0, 1, 9, 100, 966, 7284, 37476, 111064, 142773, 44947, 719, 0

Offset: 0

Alois P. Heinz, May 09 2014

From Vaclav Kotesovec, Aug 26 2014: (Start)
Column k > 0 is asymptotic to c(k) * d(k)^n / n^(3/2), where constants c(k) and d(k) are dependent only on k. Conjecture: d(k) ~ k * exp(1). Numerically:
d(1)   =   2.9557652856519949747148175... (A051491)
d(2)   =   5.6465426162329497128927135... (A245870)
d(3)   =   8.3560268792959953682760695...
d(4)   =  11.0699628777593263124193026...
d(5)   =  13.7856511100846851989303249...
d(6)   =  16.5022088446930015657112211...
d(7)   =  19.2192613290638657575973462...
d(8)   =  21.9366222112987115910888213...
d(9)   =  24.6541883249893084812976812...
d(10)  =  27.3718979186642404090999595...
d(100) = 272.0126359583480733207362718...
d(101) = 274.7309127032967881125015217...
d(200) = 543.8405620978790523837823296...
d(201) = 546.5588426492458787468860222...
d(101)-d(100) = 2.718276744...
d(201)-d(200) = 2.718280551...
(End)

			Square array A(n,k) begins:
  0,  0,    0,     0,      0,      0,       0,       0, ...
  1,  1,    1,     1,      1,      1,       1,       1, ...
  0,  1,    2,     3,      4,      5,       6,       7, ...
  0,  2,    7,    15,     26,     40,      57,      77, ...
  0,  4,   26,    82,    188,    360,     614,     966, ...
  0,  9,  107,   495,   1499,   3570,    7284,   13342, ...
  0, 20,  458,  3144,  12628,  37476,   91566,  195384, ...
  0, 48, 2058, 20875, 111064, 410490, 1200705, 2984142, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 (2018), doi SIGMA 17 (2021)

Columns k=0-10 give: A063524, A000081, A000151, A006964, A052763, A052788, A246235, A246236, A246237, A246238, A246239.
Rows n=0-3 give: A000004, A000012, A001477, A005449.
Lower diagonal gives A242375.
Cf. A255517, A256064.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
      A(d, k), d=divisors(j))*A(n-j, k)*k, j=1..n-1))/(n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

nn = 10; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; Transpose[ Table[Flatten[ sol = SolveAlways[ 0 == Series[ t[x] - x Product[1/(1 - x^i)^(n a[i]), {i, 1, nn}], {x, 0, nn}], x]; Flatten[{0, Table[a[n], {n, 1, nn}]}] /. sol], {n, 0, nn}]] // Grid (* Geoffrey Critzer, Nov 11 2014 *)
A[n_, k_] := A[n, k] = If[n<2, n, Sum[Sum[d*A[d, k], {d, Divisors[j]}] *A[n-j, k]*k, {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 04 2014, translated from Maple *)

\\ ColGf gives column generating function
ColGf(N,k) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = k/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); x*Ser(A)}
Mat(vector(8, k, concat(0, Col(ColGf(7, k-1))))) \\ Andrew Howroyd, May 12 2018

G.f. for column k: x*Product_{n>=1} 1/(1 - x^n)^(k*A(n,k)). - Geoffrey Critzer, Nov 13 2014

A005753 Number of rooted identity matched trees with n nodes.

Original entry on oeis.org

1, 2, 5, 18, 66, 266, 1111, 4792, 21124, 94888, 432415, 1994828, 9296712, 43706722, 207030398, 987130456, 4733961435, 22819241034, 110500644857, 537295738556, 2622248720234, 12840953621208, 63074566121245, 310693364823376, 1534374047239554, 7595642577152762

Offset: 1

N. J. A. Sloane

Also number of rooted identity trees with n nodes and 2-colored non-root nodes. - Christian G. Bower, Apr 15 1998

			G.f.: A(x) = x + 2*x^2 + 5*x^3 + 18*x^4 + 66*x^5 + 266*x^6 + ...
where A(x) = x*(1+x)^2*(1+x^2)^4*(1+x^3)^10*(1+x^4)^36*(1+x^5)^132*... (the exponents are A038077(n), n>=1).

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..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 429
R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1991), 93-104.
R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A038077, A246312.

Column k=2 of A255517.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2*a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n=1, 1, b((n-1)$2)):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 01 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[2*a[i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n == 1, 1, b[n-1, n-1]]; Table[a[n] // FullSimplify, {n, 1, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

{a(n)=polcoeff(x*prod(k=1, n-1, (1+x^k+x*O(x^n))^(2*a(k))), n)} /* Paul D. Hanna */

G.f.: x*Product_{n>=1} (1 + x^n)^(2*a(n)) = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Dec 31 2011
a(n) ~ c * d^n / n^(3/2), where d = A246312 = 5.249032491228170579164952216..., c = 0.192066288645200371237879149260484794708740197522264442948290580404909605849... - Vaclav Kotesovec, Aug 25 2014, updated Dec 26 2020
G.f. A(x) satisfies: A(x) = x*exp(2*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k). - Ilya Gutkovskiy, Apr 13 2019

A319254 Array read by antidiagonals: T(n,k) is the number of series-reduced rooted trees with n leaves of k colors.

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 4, 6, 10, 5, 5, 10, 28, 40, 12, 6, 15, 60, 156, 170, 33, 7, 21, 110, 430, 948, 785, 90, 8, 28, 182, 965, 3396, 6206, 3770, 261, 9, 36, 280, 1890, 9376, 28818, 42504, 18805, 766, 10, 45, 408, 3360, 21798, 97775, 256172, 301548, 96180, 2312

Offset: 1

Andrew Howroyd, Sep 15 2018

Not all colors need to be used.

See table 2.3 in the Johnson reference.

			Array begins:
==================================================================
n\k|   1     2       3        4         5         6          7
---+--------------------------------------------------------------
1  |   1     2       3        4         5         6          7 ...
2  |   1     3       6       10        15        21         28 ...
3  |   2    10      28       60       110       182        280 ...
4  |   5    40     156      430       965      1890       3360 ...
5  |  12   170     948     3396      9376     21798      44856 ...
6  |  33   785    6206    28818     97775    269675     642124 ...
7  |  90  3770   42504   256172   1068450   3496326    9632960 ...
8  | 261 18805  301548  2357138  12081605  46897359  149491104 ...
9  | 766 96180 2195100 22253672 140160650 645338444 2379859608 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012.

Columns 1..5 are A000669, A050381, A220823, A220824, A220825.
Main diagonal is A319369.
Cf. A141610, A242249, A255517, A256064, A256068, A319376.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 17 2018

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n k, b[n, n - 1, k]];
Table[A[n, 1 + d - n], {d, 1, 12}, {n, 1, d}] // Flatten (* Jean-François Alcover, Sep 11 2019, after Alois P. Heinz *)

\\ here R(n,k) gives k'th column as a vector.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v,[0]))[n])); v}
{my(T=Mat(vector(8, k, R(8, k)~))); for(n=1, #T~, print(T[n,]))} \\ Andrew Howroyd, Sep 15 2018

A052757 Number of rooted identity trees with n nodes and 3-colored non-root nodes.

Original entry on oeis.org

0, 1, 3, 12, 64, 363, 2214, 14043, 91857, 614676, 4189254, 28974915, 202870938, 1435094800, 10241197917, 73639001172, 533004547453, 3880381334415, 28395656513145, 208748382089131, 1540935621796941, 11417266889312313, 84880193073070819, 632976019285857201

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

			a(3) = 12:
o  o  o  o  o  o  o  o  o    o      o      o
|  |  |  |  |  |  |  |  |   / \    / \    / \
1  1  1  2  2  2  3  3  3  1   2  1   3  2   3
|  |  |  |  |  |  |  |  |
1  2  3  1  2  3  1  2  3  - _Alois P. Heinz_, Feb 24 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 713

Cf. A038079.

Column k=3 of A255517.

spec := [S,{S=Prod(B,B,B,Z),B=PowerSet(S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

a(n) ~ c * d^n / n^(3/2), where d = 7.969494030514425004826375511986491746399264355846412073489715938424..., c = 0.12982932099206082951153936270704832022771078... . - Vaclav Kotesovec, Feb 24 2015
From Ilya Gutkovskiy, Apr 13 2019: (Start)
G.f. A(x) satisfies: A(x) = x*exp(3*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(3*a(n)). (End)

New name from Vaclav Kotesovec, Feb 24 2015

A319376 Triangle read by rows: T(n,k) is the number of lone-child-avoiding rooted trees with n leaves of exactly k colors.

Original entry on oeis.org

1, 1, 1, 2, 6, 4, 5, 30, 51, 26, 12, 146, 474, 576, 236, 33, 719, 3950, 8572, 8060, 2752, 90, 3590, 31464, 108416, 175380, 134136, 39208, 261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032, 766, 94648, 1908858, 14047288, 49885320, 95715728, 101799712, 56555904, 12818912

Offset: 1

Andrew Howroyd, Sep 17 2018

Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.

			Triangle begins:
    1;
    1,     1;
    2,     6,      4;
    5,    30,     51,      26;
   12,   146,    474,     576,     236;
   33,   719,   3950,    8572,    8060,   2752;
   90,  3590,  31464,  108416,  175380,  134136,   39208;
  261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032;
  ...
From _Gus Wiseman_, Dec 31 2020: (Start)
The 12 trees counted by row n = 3:
  (111)    (112)    (123)
  (1(11))  (122)    (1(23))
           (1(12))  (2(13))
           (1(22))  (3(12))
           (2(11))
           (2(12))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns k=1..2 are A000669, A319377.
Main diagonal is A000311.
Row sums are A316651.
Cf. A141610, A242249, A255517, A256064, A256068, A319254, A319541, A326396.
The unlabeled version, counting inequivalent leaf-colorings of lone-child-avoiding rooted trees, is A330465.
Lone-child-avoiding rooted trees are counted by A001678 (shifted left once).
Labeled lone-child-avoiding rooted trees are counted by A060356.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000014, A000081, A000169, A005804, A206429, A330951.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
T:= (n, k)-> add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 18 2018

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n k, b[n, n - 1, k]];
T[n_, k_] := Sum[(-1)^(k - i)*Binomial[k, i]*A[n, i], {i, 1, k}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 24 2019, after Alois P. Heinz *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],1Gus Wiseman, Dec 31 2020 *)

\\ here R(n,k) is k-th column of A319254.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
M(n)={my(v=vector(n, k, R(n,k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}

T(n,k) = Sum_{i=1..k} (-1)^(k-i)*binomial(k,i)*A319254(n,i).

Sum_{k=1..n} k * T(n,k) = A326396(n). - Alois P. Heinz, Sep 11 2019

A256068 Number T(n,k) of rooted identity trees with n nodes and colored non-root nodes using exactly k colors; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 2, 14, 16, 0, 3, 60, 174, 125, 0, 6, 254, 1434, 2464, 1296, 0, 12, 1087, 10746, 33362, 40455, 16807, 0, 25, 4742, 77556, 388312, 816535, 763104, 262144, 0, 52, 21020, 551460, 4191916, 13617210, 21501684, 16328620, 4782969

Offset: 1

Alois P. Heinz, Mar 13 2015

			T(4,2) = 14:
:   0   0   0   0   0   0     0       0
:   |   |   |   |   |   |     |       |
:   1   1   2   2   2   1     1       2
:   |   |   |   |   |   |    / \     / \
:   1   2   1   2   1   2   1   2   1   2
:   |   |   |   |   |   |
:   2   1   1   1   2   1
:
:     0      0      0      0      0      0
:    / \    / \    / \    / \    / \    / \
:   1   1  2   1  1   2  2   2  1   2  2   1
:   |      |      |      |      |      |
:   2      1      1      1      2      2
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    3;
  0,  2,   14,    16;
  0,  3,   60,   174,    125;
  0,  6,  254,  1434,   2464,   1296;
  0, 12, 1087, 10746,  33362,  40455,  16807;
  0, 25, 4742, 77556, 388312, 816535, 763104, 262144;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=0-1 give: A063524 (for n>0), A004111 (for n>1):
Main diagonal gives: A000272 (for n>0).
Row sums give A319220(n-1).
T(2n+1,n) gives A309996.
Cf. A255517, A256064.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, add(A(n-j, k)*add(
      k*A(d, k)*d*(-1)^(j/d+1), d=divisors(j)), j=1..n-1)/(n-1))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n-1), n=1..10);

A[n_, k_] := A[n, k] = If[n < 2, n, Sum[A[n - j, k] Sum[k A[d, k] d * (-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n - 1}]/(n - 1)];
T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 29 2020, after Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A255517(n).

A038080 Number of identity trees with 3-colored nodes.

Original entry on oeis.org

1, 3, 3, 9, 39, 189, 981, 5490, 31674, 189954, 1170126, 7382745, 47494197, 310712808, 2061987642, 13855192866, 94113385437, 645424668666, 4464027720900, 31110200069511, 218292811705458, 1541172223659249

Offset: 0

Christian G. Bower, Jan 04 1999

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Index entries for sequences related to trees

Cf. A000220, A038077-A038079, A052757, A255517.

G.f.: B(x) - B^2(x)/2 - B(x^2)/2, where B(x) is g.f. for A038079.

a(n) ~ c * d^n / n^(5/2), where d = 7.969494030514425004826375511986491746399264355846412073489715938... and c = 0.3712461766927875417276388215355520756010680416348018056669... - Vaclav Kotesovec, Dec 26 2020

A052772 Number of rooted identity trees with n nodes and 4-colored non-root nodes.

Original entry on oeis.org

0, 1, 4, 22, 156, 1193, 9748, 82916, 727088, 6524084, 59620732, 552970626, 5191935808, 49252903050, 471358286352, 4545310993994, 44121116086052, 430777978197156, 4227634212037728, 41680927531643928, 412638233333973820, 4100336181515969163, 40882494461218775272

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 729

Column k=4 of A255517.

spec := [S,{S=Prod(Z,B,B,B,B),B=PowerSet(S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

a(n) ~ c * d^n / n^(3/2), where d = 10.68849275496965245845204879846824293047921245695819153804780100052532088..., c = 0.097992887955331161579155538221616838965194192139... . - Vaclav Kotesovec, Feb 24 2015
From Ilya Gutkovskiy, Apr 13 2019: (Start)
G.f. A(x) satisfies: A(x) = x*exp(4*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(4*a(n)). (End)

New name from Vaclav Kotesovec, Feb 24 2015

A255523 Number of rooted identity trees with n n-colored non-root nodes.

Original entry on oeis.org

1, 1, 5, 64, 1193, 30526, 991264, 39156244, 1824927697, 98125181461, 5983042467096, 408095177801851, 30797863537552547, 2548357838769171131, 229445851718471852031, 22334471403618839348901, 2337414940442888593612961, 261737726746663069945238177

Offset: 0

Alois P. Heinz, Feb 24 2015

nonn

			a(2) = 5:
  o    o    o    o      o
  |    |    |    |     / \
  1    1    2    2    1   2
  |    |    |    |
  1    2    1    2

Alois P. Heinz, Table of n, a(n) for n = 0..300

A diagonal of A255517.

Cf. A242375.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, add(A(n-j, k)*add(
      k*A(d, k)*d*(-1)^(j/d+1), d=divisors(j)), j=1..n-1)/(n-1))
    end:
a:= n-> A(n+1, n):
seq(a(n), n=0..25);

A[n_, k_] := A[n, k] = If[n < 2, n, Sum[A[n-j, k]*Sum[
     k*A[d, k]*d*(-1)^(j/d+1), {d, Divisors[j]}], {j, 1, n-1}]/(n-1)];
a[n_] := A[n+1, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz *)

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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A242249 Number A(n,k) of rooted trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A005753 Number of rooted identity matched trees with n nodes.

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A319254 Array read by antidiagonals: T(n,k) is the number of series-reduced rooted trees with n leaves of k colors.

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A052757 Number of rooted identity trees with n nodes and 3-colored non-root nodes.

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A319376 Triangle read by rows: T(n,k) is the number of lone-child-avoiding rooted trees with n leaves of exactly k colors.

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