A048160 -id:A048160 - 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

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

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

A054589 Table related to labeled rooted trees, cycles and binary trees.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 6, 18, 25, 15, 24, 96, 190, 210, 105, 120, 600, 1526, 2380, 2205, 945, 720, 4320, 13356, 26488, 34650, 27720, 10395, 5040, 35280, 128052, 305620, 507430, 575190, 405405, 135135

Offset: 1

F. Chapoton, Apr 14 2000

The left column is (n-1)!, the right column is (2n-3)!!, the total of each row is n^(n-1).
Constant terms of polynomials related to Ramanujan psi polynomials (see Zeng reference).
From Peter Bala, Sep 29 2011: (Start)
Differentiating n times the Lambert function W(x) = Sum_{n>=1} n^(n-1)*x^n/n! with respect to x yields (d/dx)^n W(x) = exp(n*W(x))/(1-W(x))^n*R(n,1/(1-W(x))), where R(n,x) is the n-th row polynomial of this triangle. The first few values are R(1,x) = 1, R(2,x) = 1+x, R(3,x) = 2+4*x+3*x^2. The Ramanujan polynomials R(n,x) are strongly x-log-convex [Chen et al.].
Shor and Dumont-Ramamonjisoa have proved independently that the coefficient of x^k in R(n,x) counts rooted labeled trees on n vertices with k improper edges. Drake, Example 1.7.3, gives another combinatorial interpretation for this triangle as counting a family of labeled trees.
(End)

			Triangle begins:
  {1},
  {1,  1},
  {2,  4,  3},
  {6, 18, 25, 15},
  ...

J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013-2014.
William Y. C. Chen, Amy M. Fu, and Elena L. Wang, A Grammatical Calculus for the Ramanujan Polynomials, arXiv:2506.01649 [math.CO], 2025. See p. 3.
William Y. C. Chen, Larry X. W. Wang, and Arthur L. B. Yang, Recurrence Relations for Strongly q-Log-Convex Polynomials, arXiv:0806.3641v1 [math.CO], 2008.
Diego Dominici, Nested derivatives: A simple method for Computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
Dominique Dumont and Armand Ramamonjisoa, Grammaire de Ramanujan et Arbres de Cayley, Electr. J. Combinatorics, Volume 3, Issue 2 (1996) R17 (see page 16).
D. J. Jeffrey, G. A. Kalugin, and N. Murdoch, Lagrange inversion and Lambert W, Preprint 2015.
Matthieu Josuat-Vergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
Peter W. Shor, A new proof of Cayley's formula for counting labeled trees, J. Combin. Theory Ser. A 71 (1995), no. 1, 154-158.
Jiang Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan J., 3(1999) 1, 45-54.

Cf. A000169, A001147, A075856, A048159, A048160, A042977.

p[1] = 1; p[n_] := p[n] = Expand[x^2*D[p[n-1], x] + (n-1)(1+x)p[n-1]]; Flatten[ Table[ CoefficientList[ p[n], x], {n, 1, 8}]] (* Jean-François Alcover, Jul 22 2011 *)
Clear[a];
a[1, 0] = 1;
a[n_, k_] /; k < 0 || k >= n := 0
a[n_, k_] /; 0 <= k <= n - 1 :=
a[n, k] = (n - 1) a[n - 1, k] + (n + k - 2) a[n - 1, k - 1]
Table[a[n, k], {n, 20}, {k, 0, n - 1}] (* David Callan, Oct 14 2012 *)

The polynomials p_n = Sum a[n, k]x^k satisfy p_1=1 and p_(n+1) = x*x*dp_n/dx+n*(1+x)*p_n.
From Peter Bala, Sep 29 2011: (Start)
E.g.f.: series reversion with respect to x of (1-t+(t-1+x*t)*exp(-x)) = x + (1+t)*x^2/2! + (2+4*t+3*t^2)*x^3/3! + ....
The sequence of shifted row polynomials {p_n(1+t)}n>=1 begins [1,2+t,9+10*t+3*t^2,...]. These are the row polynomials of A048160.
(End)
Let f(x) = exp(x)/(1-t*x). The e.g.f. A(x,t) = x + (1+t)*x^2/2! + (2+4*t+3*t^2)*x^3/3! + ... satisfies the autonomous differential equation dA/dx = f(A). The n-th row polynomial (n>=1) equals D^(n-1)(f(x)) evaluated at x = 0, where D is the operator f(x)*d/dx (apply [Dominici, Theorem 4.1]). - Peter Bala, Nov 09 2011
The polynomials (1+t)^(n-1)*p_n(1/(1+t)) are (up to sign) the row polynomials of A042977. - Peter Bala, Jul 23 2012
Let q_n = Sum_{k>=0} a(n,k)*t^(n-k), with q_0 = 1. (So q_1=t, q_2 = t+t^2, and q_3 = 3*t + 4*t^2 + 2*t^3.)  Then Sum_{n>=0} q_n*x^n/n! = t - W((t-1-t^2*x)*exp(t-1)), where W is the Lambert function. - Ira M. Gessel, Jan 06 2012

A176824 a(n) = (n+1)^n mod n^n.

Original entry on oeis.org

0, 1, 10, 113, 1526, 24337, 450066, 9492289, 225159022, 5937424601, 172385029466, 5465884225969, 187964560069638, 6968912374274593, 277133723845128226, 11767703728247765249, 531431035966023003614, 25434534147318166381993, 1286040688679372821752042

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 26 2010

G. C. Greubel, Table of n, a(n) for n = 1..380

Cf. A000169, A000312, A048160.

[(n+1)^n mod n^n: n in [1..20]]; // Vincenzo Librandi, Sep 07 2015

A176824:=n->(n+1)^n mod n^n: seq(A176824(n), n=1..25); # Wesley Ivan Hurt, Sep 10 2015

Mathematica
```
Table[Mod[(n+1)^n, n^n], {n,30}]
```

first(m)=vector(m,i,((i+1)^i) % (i^i)) \\ Anders Hellström, Sep 07 2015

[(n+1)^n%n^n for n in range(1,31)] # G. C. Greubel, May 23 2023

From Peter Bala, Sep 12 2012: (Start)
a(n) = (n+1)^n - 2*n^n (since 2*n^n <= (n+1)^n < 3*n^n for n >= 1).
In terms of the tree function T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! of A000169 the e.g.f. is T(x)*(2*x + T(x)*(T(x)-2))/(x^2*(T(x)-1)^3) = x + 10*x^2/2! + 113*x^3/3! + ... . (End)
a(n) = Sum_{i=1..n-1} C(n,i-1)*i^(i-1)*(n-i)^(n-i). - Vladimir Kruchinin, Sep 07 2015
a(n) = A000169(n+1) - 2*A000312(n). - Michel Marcus, Sep 07 2015, after Peter Bala

a(19) from Vincenzo Librandi, Sep 07 2015

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

A005264 Number of labeled rooted Greg trees with n nodes.

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A005263 Number of labeled Greg trees.

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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.

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A052303 Number of asymmetric Greg trees.

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A052301 Number of asymmetric rooted Greg trees.

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A054589 Table related to labeled rooted trees, cycles and binary trees.

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