A005263 -id:A005263 - cp's OEIS frontend

A005264 Number of labeled rooted Greg trees with n nodes.

1, 3, 22, 262, 4336, 91984, 2381408, 72800928, 2566606784, 102515201984, 4575271116032, 225649908491264, 12187240730230528, 715392567595403520, 45349581052869924352, 3087516727770990992896, 224691760916830871873536

Offset: 1

N. J. A. Sloane

A rooted Greg tree can be described as a rooted tree with 2-colored nodes where only the black nodes are counted and labeled and the white nodes have at least 2 children. - Christian G. Bower, Nov 15 1999

			G.f. = x + 3*x^2 + 22*x^3 + 262*x^4 + 4336*x^5 + 91984*x^6 + 2381408*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 1..358
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
J. Felsenstein, The number of evolutionary trees, Systematic Zoology, 27 (1978), 27-33.
J. Felsenstein, The number of evolutionary trees, Systematic Zoology, 27 (1978), 27-33. (Annotated scanned copy)
C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128.
C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128. (Annotated scanned copy)
C. Flight, Letter to N. J. A. Sloane, Nov 1990
L. R. Foulds and R. W. Robinson, Determining the asymptotic number of phylogenetic trees, pp. 110-126 of Combinatorial Mathematics VII (Newcastle, August 1979), ed. R. W. Robinson, G. W. Southern and W. D. Wallis. Lect. Notes in Math., 829. Springer, 1980.
L. R. Foulds & R. W. Robinson, Determining the asymptotic number of phylogenetic trees, Lecture Notes in Math., 829 (1980), 110-126. (Annotated scanned copy)
Armin Hoenen, Steffen Eger, and Ralf Gehrke, How Many Stemmata with Root Degree k?, Proceedings of the 15th Meeting on the Mathematics of Language, pp. 11-21, 2017.
D. J. Jeffrey, G. A. Kalugin, and N. Murdoch, Lagrange inversion and Lambert W, Preprint, 2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC).
M. Josuat-Vergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
Paul Laubie, Combinatorics of pre-Lie products sharing a Lie bracket, arXiv:2309.05552 [math.QA], 2023. See pp. 1, 5.
N. J. A. Sloane, Transforms
N. S. Wedd, Letter to N. J. A. Sloane, N.D.
Index entries for reversions of series
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Inverse Stirling transform of A005172 (hence corrected and extended). - John W. Layman

Cf. A005263, A048159, A048160, A052300, A052301, A052302, A052303, A157142.

T := proc(n,k) option remember; if k=0 and (n=0 or n=1) then return(1) fi; if k<0 or k>n then return(0) fi;
(n-1)*T(n-1,k-1)+(3*n-k-4)*T(n-1,k)-(k+1)*T(n-1,k+1) end:
A005264 := proc(n) add(T(n,k)*(-1)^k*2^(n-k-1), k=0..n-1) end;
seq(A005264(n),n=1..17); # Peter Luschny, Nov 10 2012

max = 17; f[x_] := -1/2 - ProductLog[-E^(-1/2)*(x + 1)/2]; Rest[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!] (* Jean-François Alcover, May 23 2012, after Peter Bala *)
a[ n_] := If[ n < 1, 0, n! SeriesCoefficient[ InverseSeries[ Series[ Exp[-x] (1 + 2 x) - 1, {x, 0, n}]], n]]; (* Michael Somos, Jun 07 2012 *)

a(n):=if n=1 then 1 else sum((n+k-1)!*sum(1/(k-j)!*sum(1/(l!*(j-l)!)*sum(((-1)^(i+l)*l^i*binomial(l,n+j-i-1)*2^(n+j-i-1))/i!,i,0,n+j-1),l,1,j),j,1,k),k,1,n-1); /* Vladimir Kruchinin, May 04 2012 */

{a(n) = local(A); if( n<1, 0, for( k= 1, n, A += x * O(x^k); A = truncate( (1 + x) * exp(A) - 1 - A) ); n! * polcoeff( A, n))}; /* Michael Somos, Apr 02 2007 */

{a(n) = if( n<1, 0, n! * polcoeff( serreverse( exp( -x + x * O(x^n) ) * (1 + 2*x) - 1), n))}; /* Michael Somos, Mar 26 2011 */

@CachedFunction
def T(n,k):
    if k==0 and (n==0 or n==1): return 1
    if k<0 or k>n: return 0
    return (n-1)*T(n-1,k-1)+(3*n-k-4)*T(n-1,k)-(k+1)*T(n-1,k+1)
A005264 = lambda n: add(T(n,k)*(-1)^k*2^(n-k-1) for k in (0..n-1))
[A005264(n) for n in (1..17)]  # Peter Luschny, Nov 10 2012

Exponential reversion of A157142 with offset 1. - Michael Somos, Mar 26 2011
The REVEGF transform of the odd numbers [1,3,5,7,9,11,...] is  [1, -3, 22, -262, 4336, -91984, 2381408, ...] - N. J. A. Sloane, May 26 2017
E.g.f. A(x) = y satisfies y' = (1 + 2*y) / ((1 - 2*y) * (1 + x)). - Michael Somos, Mar 26 2011
E.g.f. A(x) satisfies (1 + x) * exp(A(x)) = 1 + 2 * A(x).
From Peter Bala, Sep 08 2011: (Start)
A(x) satisfies the separable differential equation A'(x) = exp(A(x))/(1-2*A(x)) with A(0) = 0. Thus the inverse function A^-1(x) = int {t = 0..x} (1-2*t)/exp(t) = exp(-x)*(2*x+1)-1 = x-3*x^2/2!+5*x^3/3!-7*x^4/4!+.... A(x) = -1/2-LambertW(-exp(-1/2)*(x+1)/2).
The expansion of A(x) can be found by inverting the above integral using the method of [Dominici, Theorem 4.1] to arrive at the result a(n) = D^(n-1)(1) evaluated at x = 0, where D denotes the operator g(x) -> d/dx(exp(x)/(1-2*x)*g(x)). Compare with [Dominici, Example 9].
(End)
a(n)=sum(k=1..n-1, (n+k-1)!*sum(j=1..k, 1/(k-j)!*sum(l=1..j,  1/(l!*(j-l)!)* sum(i=0..n+j-1, ((-1)^(i+l)*l^i*binomial(l,n+j-i-1)*2^(n+j-i-1))/i!)))), n>1, a(1)=1. - Vladimir Kruchinin, May 04 2012
Let T(n,k) = 1 if k=0 and (n=0 or n=1); T(n,k) = 0 if k<0 or k>n; and otherwise T(n,k) = (n-1)*T(n-1,k-1)+(3*n-k-4)*T(n-1,k)-(k+1)*T(n-1,k+1). Define polynomials p(n,w) = w^n*sum_{k=0..n-1}(T(n,k)*w^k)/(1+w)^(2*n-1), then a(n) = (-1)^n*p(n,-1/2). - Peter Luschny, Nov 10 2012
a(n) ~ n^(n-1) / (sqrt(2) * exp(n/2) * (2-exp(1/2))^(n-1/2)). - Vaclav Kotesovec, Jul 09 2013
E.g.f.: -W(-(1+x)*exp(-1/2)/2)-1/2 where W is the Lambert W function. - Robert Israel, Mar 28 2017

A048160 Triangle giving T(n,k) = number of (n,k) labeled rooted Greg trees (n >= 1, 0<=k<=n-1).

Original entry on oeis.org

1, 2, 1, 9, 10, 3, 64, 113, 70, 15, 625, 1526, 1450, 630, 105, 7776, 24337, 31346, 20650, 6930, 945, 117649, 450066, 733845, 650188, 329175, 90090, 10395, 2097152, 9492289, 18760302, 20925065, 14194180, 5845455, 1351350, 135135, 43046721

Offset: 1

N. J. A. Sloane

An (n,k) rooted Greg tree can be described as a rooted tree with n black nodes and k white nodes where only the black nodes are labeled and the white nodes have at least 2 children. - Christian G. Bower, Nov 15 1999

			Triangle begins:
  1;
  2, 1;
  9, 10, 3;
  64, 113, 70, 15;
  ...

C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128.
C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128. (Annotated scanned copy)
C. Flight, Letter to N. J. A. Sloane, Nov 1990
D. J. Jeffrey, G. A. Kalugin, N. Murdoch, Lagrange inversion and Lambert W, Preprint 2015.
M. Josuat-Vergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Row sums give A005264. Cf. A005263, A048159, A052300-A052303. A054589.

t[n_ /; n >= 1, k_ /; k >= 0] /; 0 <= k <= n-1 := t[n, k] = (n+k-2) t[n-1, k-1] + (2n + 2k - 2)*t[n-1, k] + (k+1) t[n-1, k+1]; t[1, 0] = 1; t[, ] = 0; Flatten[Table[t[n, k], {n, 1, 9}, {k, 0, n-1}]] (* Jean-François Alcover, Jul 20 2011, after formula *)

T(n, 0)=n^(n-1), T(n, k)=(n+k-2)*T(n-1, k-1)+(2*n+2*k-2)*T(n-1, k)+(k+1)*T(n-1, k+1).
From Peter Bala, Sep 29 2011: (Start)
E.g.f.: compositional inverse with respect to x of t*(exp(-x)-1) + (1+t)*x*exp(-x) = compositional inverse with respect to x of (x - (2+t)*x^2/2! + (3+2*t)*x^3/3! - (4+3*t)*x^4/4! + ...) = x + (2+t)*x^2/2! + (9+10*t+3*t^2)*x^3/3! + ....
The row generating polynomials R(n,t) satisfy the recurrence R(n+1,t) = (1+t)^2*R'(n,t)+n*(2+t)*R(n,t) with R(1,t) = 1.
The shifted row polynomials R(n,t-1) are the row generating polynomials of A054589. (End)
From Peter Bala, Sep 12 2012: (Start)
It appears that the entries in column k = 1 are given by T(n,1) = (n+1)^n - 2*n^n (checked up to n = 15) - see A176824.
Assuming this, we could then use the recurrence equation to obtain explicit formulas for columns k = 2,3,....
For example, T(n,2) = 1/2*{(n+2)^(n+1) - 4*(n+1)^(n+1) + (4*n+3)*n^n}. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000

A048159 Triangle giving a(n,k) = number of (n,k) labeled Greg trees (n >= 2, 0 <= k <= n-2).

Original entry on oeis.org

1, 3, 1, 16, 13, 3, 125, 171, 85, 15, 1296, 2551, 2005, 735, 105, 16807, 43653, 47586, 26950, 7875, 945, 262144, 850809, 1195383, 924238, 412650, 100485, 10395, 4782969, 18689527, 32291463, 31818045, 19235755, 7113645, 1486485, 135135

Offset: 2

N. J. A. Sloane

An (n,k) Greg tree can be described as a tree with n black nodes and k white nodes where only the black nodes are labeled and the white nodes are of degree at least 3.

Row sums give A005263.

			Triangle begins
    1;
    3,   1;
   16,  13,   3;
  125, 171,  85,  15;
  ...

C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128.
C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128. (Annotated scanned copy)
C. Flight, Letter to N. J. A. Sloane, Nov 1990
M. Josuat-Vergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
Lucas Randazzo, Arboretum for a generalization of Ramanujan polynomials, arXiv:1905.02083 [math.CO], 2019.
Index entries for sequences related to trees

Cf. A005264, A048160, A052300, A052301, A052302, A052303.

a[n_, 0] := n^(n-2); a[n_ /; n >= 2, k_] /; 0 <= k <= n-2 := a[n, k] = (n+k-3)*a[n-1, k-1] + (2*n+2*k-3)*a[n-1, k] + (k+1)*a[n-1, k+1]; a[n_, k_] = 0; Table[a[n, k], {n, 2, 9}, {k, 0, n-2}] // Flatten (* Jean-François Alcover, Oct 03 2013 *)

a(n, 0) = n^(n-2), a(n, k) = (n+k-3)*a(n-1, k-1) + (2n+2k-3)*a(n-1, k) + (k+1)*a(n-1, k+1).

More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000

A005640 Number of phylogenetic trees with n labels.

Original entry on oeis.org

1, 1, 2, 8, 64, 832, 15104, 352256, 10037248, 337936384, 13126565888, 577818263552, 28425821618176, 1545553369366528, 92034646352592896, 5956917762776367104, 416397789920380321792, 31262503202358260924416, 2508985620606225641111552, 214348807882902869374926848, 19422044917978876510600167424

Offset: 0

N. J. A. Sloane

Each node of the tree is a subset of the labeled set {1,...,n}. If the subset node is empty, it must have degree at least 3.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.26.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
L. R. Foulds and R. W. Robinson, Determining the asymptotic number of phylogenetic trees, pp. 110-126 of Combinatorial Mathematics VII (Newcastle, August 1979), ed. R. W. Robinson, G. W. Southern and W. D. Wallis. Lecture Notes in Math., 829 (1980), 110-126. (Annotated scanned copy)
J. P. Hayes, Enumeration of fanout-free Boolean functions, J. ACM, 23 (1976), 700-709.
K. L. Kodandapani and S. C. Seth, On combinational networks with restricted fan-out, IEEE Trans. Computers, 27 (1978), 309-318. (Annotated scanned copy)
N. J. A. Sloane, Transforms
Index entries for sequences related to trees

Cf. A000311, A005172, A005263, A006351.

a[n_ /; n > 2] := 2^(n-1)*(n-2)!*Sum[ Binomial[n+k-2, n-2]*Sum[ (-1)^j*Binomial[k, j]*Sum[ ((-1)^l*2^(j-l)*Binomial[j, l]*(j-l)!*StirlingS1[n+j-l-2, j-l])/(n+j-l-2)!, {l, 0, j}], {j, 1, k}], {k, 1, n-2}]; a[0] = a[1] = 1; a[2] = 2; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 10 2012, after Vladimir Kruchinin *)

STIRLING transform of A005263.
E.g.f.: 1+B(x)-B(x)^2 where B(x) is e.g.f. of A005172.
For n >= 2, a(n) = 2^n*A006351(n) = 2^(n+1)*A000311(n).

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

A052300 Number of rooted Greg trees.

Original entry on oeis.org

1, 2, 6, 21, 78, 313, 1306, 5653, 25088, 113685, 523522, 2443590, 11533010, 54949539, 263933658, 1276652682, 6213207330, 30402727854, 149486487326, 738184395770, 3659440942282, 18205043615467, 90856842218506, 454770531433586, 2282393627458496, 11483114908752959

Offset: 1

Christian G. Bower, Nov 15 1999

A rooted Greg tree can be described as a rooted tree with 2-colored nodes where only the black nodes are counted and the white nodes have at least 2 children.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to rooted trees
N. J. A. Sloane, Transforms

Cf. A005263, A005264, A048159, A048160, A052301-A052303.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<1, 0, b(n-1$2)+b(n, n-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 22 2018

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i] + j - 1, j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := If[n < 1, 0, b[n - 1, n - 1] + b[n, n - 1]];
a /@ Range[1, 40] (* Jean-François Alcover, Oct 02 2019, after Alois P. Heinz *)

Satisfies a = EULER(a) + SHIFT_RIGHT(EULER(a)) - a.

a(n) ~ c * d^n / n^(3/2), where d = 5.33997181362574740496306748840603859910694551382103293340704... and c = 0.18146848896221859476228524468003196434835879494225205... - Vaclav Kotesovec, Jun 11 2021

A052303 Number of asymmetric Greg trees.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 4, 12, 42, 137, 452, 1491, 4994, 16831, 57408, 197400, 685008, 2395310, 8437830, 29917709, 106724174, 382807427, 1380058180, 4998370015, 18181067670, 66393725289, 243347195594, 894959868983, 3301849331598, 12217869541117, 45335177297876

Offset: 0

Christian G. Bower, Nov 15 1999

nonn

A Greg tree can be described as a tree with 2-colored nodes where only the black nodes are counted and the white nodes are of degree at least 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1668
Index entries for sequences related to trees

Cf. A005263, A005264, A048159, A048160, A052300-A052302.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
g:= n-> `if`(n<1, 0, b(n-1$2)+b(n, n-1)) :
a:= n-> `if`(n=0, 1, g(n)-add(g(j)*g(n-j), j=0..n)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 22 2018

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
g[n_] := If[n < 1, 0, b[n - 1, n - 1] + b[n, n - 1]];
a[n_] := If[n == 0, 1, g[n] - Sum[g[j] g[n - j], {j, 0, n}]];
a /@ Range[0, 40] (* Jean-François Alcover, Apr 28 2020, after Alois P. Heinz *)

G.f.: 1+B(x)-B(x)^2 where B(x) is g.f. of A052301.

A052301 Number of asymmetric rooted Greg trees.

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 138, 455, 1540, 5305, 18546, 65616, 234546, 845683, 3072350, 11235393, 41326470, 152793376, 567518950, 2116666670, 7924062430, 29765741831, 112157686170, 423809991041, 1605622028100, 6097575361683, 23207825593664, 88512641860558

Offset: 1

Christian G. Bower, Nov 15 1999

A rooted Greg tree can be described as a rooted tree with 2-colored nodes where only the black nodes are counted and the white nodes have at least 2 children.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Essentially the same as A031148. Cf. A005263, A005264, A048159, A048160, A052300-A052303.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<1, 1, b(n-1$2)) +b(n, n-1):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 06 2014

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n<1, 1, b[n-1, n-1]] + b[n, n-1];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Satisfies a = WEIGH(a) + SHIFT_RIGHT(WEIGH(a)) - a.

a(n) ~ c * d^n / n^(3/2), where d = 4.0278584853545190803008179085023154..., c = 0.14959176868229550510957320468... . - Vaclav Kotesovec, Sep 12 2014

A052302 Number of Greg trees with n black nodes.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 37, 116, 412, 1526, 5995, 24284, 101619, 434402, 1893983, 8385952, 37637803, 170871486, 783611214, 3625508762, 16906577279, 79395295122, 375217952457, 1783447124452, 8521191260092, 40907997006020, 197248252895597, 954915026282162

Offset: 0

Christian G. Bower, Nov 15 1999

nonn

A Greg tree can be described as a tree with 2-colored nodes where only the black nodes are counted and the white nodes are of degree at least 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to trees

Cf. A005263, A005264, A048159, A048160, A052300, A052301, A052303.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
g:= n-> `if`(n<1, 0, b(n-1$2)+b(n, n-1)):
a:= n-> `if`(n=0, 1, g(n)-add(g(j)*g(n-j), j=0..n)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 22 2018

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
    Sum[Binomial[g[i] + j - 1, j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
g[n_] := If[n < 1, 0, b[n - 1, n - 1] + b[n, n - 1]];
a[n_] := If[n == 0, 1, g[n] - Sum[g[j]*g[n - j], {j, 0, n}]];
a /@ Range[0, 40] (* Jean-François Alcover, Jun 11 2021, after Alois P. Heinz *)

G.f.: 1 + B(x) - B(x)^2 where B(x) is g.f. of A052300.

A286432 Numbers of labeled rooted Greg trees (A005264) with n nodes and root degree 2.

Original entry on oeis.org

0, 1, 12, 151, 2545, 54466, 1417318, 43472780, 1536228588, 61466251616, 2746907348768, 135619260805568, 7331022129923648, 430638151053316480, 27315015477709844352, 1860627613021322933248, 135465573609158928964096, 10498038569346091127451136, 862792664850194915870874112

Offset: 1

Armin Hoenen, May 09 2017

nonn

Numbers of rooted Greg trees with 2 subtrees below root given m labeled nodes (lead index). Among all trees at the same index (see sequence A005264) root bifurcating trees play a central role in philological discourse on the reconstruction of manuscript genealogies. Labeled nodes represent surviving manuscripts, unlabeled nodes hypothetical ones. See also stemmatology/stemmatics, Bédier's paradox.

			For n=3, T_{3,2} is T_{3,0,2} + T_{3,1,2} + T_{3,2,2} where T_{3,0,2} = (3/2) * (binomial(2,(1,1)) * product(g(1,0)*g(1,0))) + 0 = 3; T_{3,1,2} = 0 + 1/2 * ((binomial(3,(2,1)) * product(g(2,0)*g(1,0))) + (binomial(3,(1,2)) * product(g(1,0)*g(2,0)))) = 6 and T_{3,2,2} = 0 + (1/2) * ((binomial(3,(2,1)) * product(g(2,1)*g(1,0))) + (binomial(3,(1,2)) * product(g(1,0)*g(2,1)))) = 3; 3 + 6 + 3 =12.

J. Bédier. La tradition manuscrite du Lai de l'Ombre: Réflexions sur l'Art d' Éditer les Anciens Textes. Romania 394 (1928), 161-196/321-356.
C. Flight. How many stemmata? Manuscripta 34(2), 1990, 122-128.
W. Hering. Zweispaltige Stemmata. Philologus-Zeitschrift für antike Literatur und ihre Rezeption 111(1-2), (1967), 170-185.
P. Maas. Textkritik. 4. Auflage. Leipzig: Teubner. 1960.

Armin Hoenen, Table of n, a(n) for n = 1..245
Armin Hoenen, S. Eger and R. Gehrke, How many stemmata with root degree k?, Proceedings of MOL 2017, 2017.

Cf. A005264, number of labeled rooted Greg trees with n nodes.

Cf. A005263, unrooted Greg trees, according to Flight (1990) can also serve as basis for computation of A005624.

T_{m,2} = Sum_{n >= 0} T_{m,n,2}, where T_{m,n,k} = (m/k!) * Sum_{(s,p) in C((m-1,n),k)} (binomial(m-1,s) F(s,p)) + (1/k!) * Sum_{(s,p) in C((m,n-1),k)} (binomial(m,s) F(s,p)), with F(s,p) = Product_{1..k} (g(s_i,p_i)), here g(m,n) = numbers of rooted Greg trees, see (A005264) with m labeled and n unlabeled nodes. s and p are tuples with k elements where each s_i >= 1 and for each p_i : 0 <= p_i < s_i; first term in T_{m,n,k} gives the number of trees with a labeled root, second those for root unlabeled.

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A266757 Erroneous version of A005263.

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A005264 Number of labeled rooted Greg trees with n nodes.

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A048160 Triangle giving T(n,k) = number of (n,k) labeled rooted Greg trees (n >= 1, 0<=k<=n-1).

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A048159 Triangle giving a(n,k) = number of (n,k) labeled Greg trees (n >= 2, 0 <= k <= n-2).

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A005640 Number of phylogenetic trees with n labels.

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A052300 Number of rooted Greg trees.

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A052303 Number of asymmetric Greg trees.

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