A005264 -id:A005264 - cp's OEIS frontend

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

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.

A157142 Signed denominators of Leibniz series for Pi/4.

Original entry on oeis.org

1, -3, 5, -7, 9, -11, 13, -15, 17, -19, 21, -23, 25, -27, 29, -31, 33, -35, 37, -39, 41, -43, 45, -47, 49, -51, 53, -55, 57, -59, 61, -63, 65, -67, 69, -71, 73, -75, 77, -79, 81, -83, 85, -87, 89, -91, 93, -95, 97, -99, 101, -103, 105, -107, 109, -111, 113, -115

Offset: 0

Jaume Oliver Lafont, Feb 24 2009

Numerators are all 1.

a(n) is also the determinant of the n X n matrix with 1's on the diagonal and 2's elsewhere (cf. A000354). - Jody Nagel (SejeongY(AT)aol.com), May 01 2010

			G.f.  = 1 - 3*x + 5*x^2 - 7*x^3 + 9*x^4 - 11*x^5 + 13*x^6 - 15*x^7 + 17*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
X. Gourdon and P. Sebah, Archimedes' constant
Mathpages, How Leibniz might have anticipated Euler
Wikipedia, Leibniz formula for Pi
Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A005264, A005408, A033999.

Cf. A157327. [Jaume Oliver Lafont, Mar 03 2009]

[(2*n + 1) * (-1)^n: n in [0..70]]; // Vincenzo Librandi, Dec 23 2018

a[ n_] := (2*n + 1) * (-1)^n; (* Michael Somos, Nov 21 2022 *)

PARI
```
{a(n) = (2*n + 1) * (-1)^n};
```

Euler transform of length 2 sequence [-3, 2]. - Michael Somos, Mar 26 2011
a(n) = b(2*n + 1) where b(n) is completely multiplicative with b(2) = 0, b(p) = p if p == 1 (mod 4), b(p) = -p if p == 3 (mod 4). - Michael Somos, Mar 26 2011
With offset 1 this sequence is the exponential reversion of A005264. - Michael Somos, Mar 26 2011
a(-1 - n) = a(n),  a(n + 1) + a(n - 1) = -2*a(n) for all n in Z. - Michael Somos, Mar 26 2011
E.g.f.: (1 - 2*x)*exp(-x). - Michael Somos, Mar 26 2011
a(n) = A005408(n)*A033999(n).
G.f.: (1 - x)/(1 + x)^2 = (1 - x)^3 / (1 - x^2)^2.
a(0) = 1, a(1) = -3, a(n) = -2*a(n-1) - a(n-2) for n >= 2.
Sum_{n=0..inf} 1/a(n) = Pi/4.

A005172 Number of labeled rooted trees of subsets of an n-set.

Original entry on oeis.org

1, 4, 32, 416, 7552, 176128, 5018624, 168968192, 6563282944, 288909131776, 14212910809088, 772776684683264, 46017323176296448, 2978458881388183552, 208198894960190160896, 15631251601179130462208, 1254492810303112820555776, 107174403941451434687463424, 9711022458989438255300083712

Offset: 1

N. J. A. Sloane

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

John W. Layman observes that this is the Stirling transform of A005264.

			x + 4*x^2 + 32*x^3 + 416*x^4 + 7552*x^5 + 176128*x^6 + 5018624*x^7 + ...
D^3(1) = 32*(12*x^2+28*x+13)/(2*x-1)^6. Evaluated at x = 0 this gives a(4) = 416.
From _Peter Bala_, Sep 06 2011: (Start)
a(3) = 32: The 32 increasing plane trees on 3 vertices with vertices of outdegree k coming in 2^(k+1) colors are
.
           1(x4 colors)      1(x8 colors)      1(x8 colors)
           |                / \               / \
           2(x4 colors)    2   3             3   2
           |
           3
.
  Totals: 16        +        8        +        8     = 32.   (End)

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.

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 1..240 (terms 1..30 from T. D. Noe)
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
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 and R. W. Robinson, Determining the asymptotic number of phylogenetic trees, 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.
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]
István Mezo, Victor H. Moll, José L. Ramírez, and Diego Villamizar, On the r-Derangements of type B, arXiv:2103.04151 [math.CO], 2021.
Index entries for sequences related to trees
Index entries for sequences related to rooted trees

Cf. A005640, A032188.

See A225170 for another version.

with(combinat); A005172 := n -> add(eulerian2(n-1,k)*2^(2*n-k-2),k=0..n-1): seq(A005172(n), n=1..16); # Peter Luschny, Nov 10 2012
A005172_list := proc(len) local A, n; A[1] := 1; for n from 2 to len do
A[n] := 2*A[n-1] + add(binomial(n,j)*A[j]*A[n-j], j=1..n-1) od:
convert(A,list) end: A005172_list(19); # Peter Luschny, May 24 2017

max = 16; g[x_] := -1/2 - ProductLog[-E^(-1/2 + x)/2]; Drop[ CoefficientList[ Series[ g[x], {x, 0, max}], x]*Range[0, max]!, 1](* Jean-François Alcover, Nov 17 2011, after 1st e.g.f. *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ -1/2 - ProductLog[ -Exp[ -1/2 + z] / 2], {z, 0, n}]] (* Michael Somos, Jun 07 2012 *)
a[1] = 1; a[n_] := (Sum[(n + k - 1)!*Sum[(-1)^j/(k - j)!*Sum[(-1)^i*2^(n - i + j - 1)*StirlingS1[n - i + j - 1, j - i]/((n - i + j - 1)!*i!), {i, 0, j}], {j, 1, k}], {k, 1, n - 1}]);
Array[a, 20] (* Jean-François Alcover, Jun 24 2018, after Vladimir Kruchinin *)
Eulerian2[n_, k_] := Eulerian2[n, k] = If[k == 0, 1, If[k == n, 0, Eulerian2[n-1, k] (k + 1) + Eulerian2[n - 1, k - 1] (2n - k - 1)]]; A005172[n_] := Sum[Eulerian2[n - 1, k] 2^(2 n - k - 2), {k, 0, n - 1}]; Table[A005172[n], {n, 19}] (* Peter Luschny, Jun 24 2018 *)

a(n):=if n=1 then 1 else (sum((n+k-1)!*sum((-1)^j/(k-j)!*sum((-1)^i*2^(n-i+j-1)*stirling1(n-i+j-1,j-i)/((n-i+j-1)!*i!),i,0,j),j,1,k),k,1,n-1)); /* Vladimir Kruchinin, Jan 30 2012 */

{a(n)=local(A=1+x);for(i=0,n,A=1+intformal(A*(1+A+x*O(x^n))^2));n!*polcoeff(A,n)} \\ Paul D. Hanna, Sep 06 2008

N=66; x='x+O('x^N);
Q(k)=if(k>N,1,1-2*(k+1)*x-2*x*(k+1)/Q(k+1));
gf=1/Q(0);  Vec(gf) \\ Joerg Arndt, May 01 2013

@CachedFunction
def eulerian2(n, k):
    if k==0: return 1
    elif k==n: return 0
    return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
A005172 = lambda n: add(eulerian2(n-1,k)*2^(2*n-k-2) for k in (0..n-1))
[A005172(n) for n in (1..16)]  # Peter Luschny, Nov 10 2012

E.g.f.: -1/2 - LambertW ( - exp( -1/2 + x) / 2 ).
E.g.f.: A(x) = 1 + Integral A(x)*(1 + A(x))^2 dx. - Paul D. Hanna, Sep 06 2008
From Peter Bala, Sep 06 2011: (Start)
The generating function A(x) = x+4*x^2/2!+32*x^3/3!+... satisfies the autonomous differential equation A'(x) = (1+2*A)/(1-2*A) with A(0) = 0. Hence the inverse function A^-1(x) = Integral_{t = 0..x} (1-2*t)/(1+2*t) dt = log(1+2*x)-x.
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((1+2*x)/(1-2*x)*g(x)). Compare with A032188.
Applying [Bergeron et al., Theorem 1] to the result x = Integral_{t = 0..A(x)} 1/phi(t) dt, where phi(t) = (1+2*t)/(1-2*t) = 1+4*t+8*t^2+16*t^3+32*t^4+... leads to the following combinatorial interpretation for this sequence: a(n) gives the number of plane increasing trees on n vertices where each vertex of outdegree k >= 1 can be colored in 2^(k+1) ways. An example is given below. (End)
a(n) = Sum_{k=1..n-1} (n+k-1)!*Sum_{j=1..k} (-1)^j/(k-j)!*Sum_{i=0..j} (-1)^i* 2^(n-i+j-1)*Stirling1(n-i+j-1,j-i)/((n-i+j-1)!*i!), n>1, a(1)=1. - Vladimir Kruchinin, Jan 30 2012
Let p(n,w) = w*Sum_{k=0..n-1}((-1)^k*E2(n-1,k)*w^k)/(1+w)^(2*n-1), E2 the second-order Eulerian numbers as defined by Knuth, then a(n) = -p(n,-1/2). - Peter Luschny, Nov 10 2012
a(n) ~ (2/(2*log(2)-1))^(n-1/2)*n^(n-1)/exp(n). - Vaclav Kotesovec, Jan 05 2013
G.f.: 1/Q(0), where Q(k)= 1 - 2*(k+1)*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
a(n) = 2*a(n-1) + Sum_{j=1..n-1} binomial(n,j)*a(j)*a(n-j) for n>1, a(1) = 1. - Peter Luschny, May 24 2017
a(1) = 1; a(n) = n! * [x^n] exp(x + Sum_{k=1..n-1} a(k)*x^k/k!). - Ilya Gutkovskiy, Oct 18 2017

A005263 Number of labeled Greg trees.

Original entry on oeis.org

1, 1, 1, 4, 32, 396, 6692, 143816, 3756104, 115553024, 4093236352, 164098040448, 7345463787136, 363154251536896, 19653476190481408, 1155636468524067328, 73364615077878838784, 5001199614295920565248, 364363128390631094137856

Offset: 0

N. J. A. Sloane

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

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 = 0..359
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 & R. W. Robinson, Determining the asymptotic number of phylogenetic trees, Lecture Notes in Math., 829 (1980), 110-126. (Annotated scanned copy)
V. Kurauskas, On graphs containing few disjoint excluded minors. Asymptotic number and structure of graphs containing few disjoint minors K_4, arXiv preprint arXiv:1504.08107 [math.CO], V1, Apr 30, 2015; V2, Jul 14 2019.
Dimitris Papamichail, Angela Huang, Edward Kennedy, Jan-Lucas Ott, Andrew Miller, Georgios Papamichail, Most Compact Parsimonious Trees, arXiv preprint arXiv:1603.03315 [cs.DS], 2016.
Index entries for sequences related to trees

Cf. A005264, A005640, A048159, A048160, A052300-A052303.

E:= 1/4 -LambertW(-(1+x)*exp(-1/2)/2)^2 - 2*LambertW(-(1+x)*exp(-1/2)/2):
S:= series(E,x,21):
seq(coeff(S,x,j)*j!, j=0..20); # Robert Israel, Mar 28 2017

max = 18; b[x] := -1/2 - ProductLog[-Exp[-1/2]*(x+1)/2]; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; sol = SolveAlways[ Normal[ Series[f[x] - (1 + b[x] - b[x]^2), {x, 0, max}]] == 0, x]; First[Table[c[k], {k, 0, max}] /. sol]*Range[0, max]! (* Jean-François Alcover, May 21 2012, from e.g.f. *)
a[ n_] := If[ n < 1, Boole[n == 0], n! SeriesCoefficient[ With[ {B =      InverseSeries[ Series[ Exp[-x] (1 + 2 x) - 1, {x, 0, n}]]}, B - B^2], n]] (* Michael Somos, Jun 07 2012 *)

{a(n) = local(A); if( n<1, n==0, for( k=1, n, A += x * O(x^k); A = truncate( (1 + x) * exp(A) - 1 - A) ); A += x * O(x^n); A -= A^2; n! * polcoeff( A, n))} /* Michael Somos, Apr 02 2007 */

E.g.f.: 1 + B(x) - B(x)^2 where B(x) is e.g.f. of A005264.
a(n) ~ n^(n-2) / (sqrt(2) * exp(n/2) * (2-exp(1/2))^(n-3/2)). - Vaclav Kotesovec, Jul 09 2013
E.g.f.: 1/4 - W(-(1+x)*exp(-1/2)/2)^2 - 2*W(-(1+x)*exp(-1/2)/2) where W is the Lambert W function. - Robert Israel, Mar 28 2017

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

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

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.

cp's OEIS Frontend

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

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A157142 Signed denominators of Leibniz series for Pi/4.

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A005172 Number of labeled rooted trees of subsets of an n-set.

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A005263 Number of labeled Greg trees.

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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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A052300 Number of rooted Greg trees.