A000891 -id:A000891 - cp's OEIS frontend

A358581 Number of rooted trees with n nodes, most of which are leaves.

1, 0, 1, 1, 4, 5, 20, 28, 110, 169, 663, 1078, 4217, 7169, 27979, 49191, 191440, 345771, 1341974, 2477719, 9589567, 18034670, 69612556, 132984290, 511987473, 991391707, 3807503552, 7460270591, 28585315026, 56595367747, 216381073935, 432396092153

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(1) = 1 through a(7) = 20 trees:
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)
                     ((ooo))  ((oooo))  ((ooooo))
                     (o(oo))  (o(ooo))  (o(oooo))
                     (oo(o))  (oo(oo))  (oo(ooo))
                              (ooo(o))  (ooo(oo))
                                        (oooo(o))
                                        (((oooo)))
                                        ((o)(ooo))
                                        ((o(ooo)))
                                        ((oo)(oo))
                                        ((oo(oo)))
                                        ((ooo(o)))
                                        (o((ooo)))
                                        (o(o)(oo))
                                        (o(o(oo)))
                                        (o(oo(o)))
                                        (oo((oo)))
                                        (oo(o)(o))
                                        (oo(o(o)))
                                        (ooo((o)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For equality we have A185650 aerated, ranked by A358578.
The opposite version is A358582, non-strict A358584.
The non-strict version is A358583.
The ordered version is A358585, odd-indexed terms A065097.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
A358589 counts square trees, ranked by A358577, ordered A358590.
Cf. A000891, A109129, A342507, A358579, A358580, A358586, A358591.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}]>Count[#,[_],{0,Infinity}]&]],{n,0,10}]

\\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[n\2+1..#u]))} \\ Andrew Howroyd, Dec 31 2022

A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=floor(n/2)+1..n} A055277(n, k). - Andrew Howroyd, Dec 31 2022

Terms a(19) and beyond from Andrew Howroyd, Dec 31 2022

A005558 a(n) is the number of n-step walks on square lattice such that 0 <= y <= x at each step.

Original entry on oeis.org

1, 1, 3, 6, 20, 50, 175, 490, 1764, 5292, 19404, 60984, 226512, 736164, 2760615, 9202050, 34763300, 118195220, 449141836, 1551580888, 5924217936, 20734762776, 79483257308, 281248448936, 1081724803600, 3863302870000, 14901311070000, 53644719852000

Offset: 0

N. J. A. Sloane

Number of n-step walks that start at the origin, constrained to stay in the first octant (0 <= y <= x). (Conjectured) - Benjamin Phillabaum, Mar 11 2011, corrected by Robert Israel, Oct 07 2015
For n >= 1, a(n-1) is the number of Dyck Paths with semilength n having floor((n+2)/2) U's in odd numbered positions. Example: (U is in odd numbered position and u is in even numbered position) Dyck path with n=5, floor ((5+2)/2)=3: UuddUuUddd. - Roger Ford, May 27 2017
The ratio of the number of n-step walks on the octant with an equal number of North steps and South steps to the total number of n-step walks on the octant is A005817(n)/a(n).  For the reduced ratio, if n is divisible by 4 or n-1 is divisible by 4 the ratio is 1:floor(n/4)+1 and for all other values of n the ratio is 2:floor(n/2)+2.  Example n = 4: A005817(4) = 10; EEEE, EEEW, EEWE, EWEE, EWEW, EEWW, ENSE, ENES, ENSW, EENS; a(4) = 20; 10:20 reduces to 1:2. - Roger Ford, Nov 04 2019

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..1000
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Slides for Séminaire de Combinatoire Ph. Flajolet, Mar 28 2013.
Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d'Informatique de Paris Nord, Université Paris 13, December 2017.
Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, and Lucien Pech, Hypergeometric expressions for generating functions of walks with small steps in the quarter plane, Eur. J. Comb. 61, 242-275 (2017)
Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2020.
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6. See Column 1 of Figure 5.
Heinrich Niederhausen, A Note on the Enumeration of Diffusion Walks in the First Octant by Their Number of Contacts with the Diagonal, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.3.

See A138350 for a signed version.
Bisections are A000891 and A000888/2.
Cf. A005559, A005560, A005561, A005562, A093768.
Cf. A000108, A005817. Column y=0 of A052174.

[Binomial(n+1, Ceiling(n/2))*Binomial(n, Floor(n/2)) - Binomial(n+1, Ceiling((n-1)/2))*Binomial(n, Floor((n-1)/2)): n in [0..30]]; // Vincenzo Librandi, Sep 30 2015

A:= proc(n,x,y) option remember;
    local j, xpyp, xp,yp, res;
    xpyp:= [[x-1,y],[x+1,y],[x,y-1],[x,y+1]];
    res:= 0;
    for j from 1 to 4 do
      xp:= xpyp[j,1];
      yp:= xpyp[j,2];
      if xp < 0 or xp > yp or xp + yp > n then next fi;
      res:= res + procname(n-1,xp,yp)
    od;
return res
end proc:
A(0,0,0) := 1:
seq(add(add(A(n,x,y), y = x .. n - x), x = 0 .. floor(n/2)), n = 0 .. 50); # Robert Israel, Oct 07 2015

a[n_] := 1/2*Binomial[2*Floor[n/2]+1, Floor[n/2]+1]*CatalanNumber[1/2*(n+Mod[n, 2])]*(Mod[n, 2]+2); Table[a[n]//Abs, {n, 0, 27}] (* Jean-François Alcover, Mar 13 2014 *)

a(n)=binomial(n+1,ceil(n/2))*binomial(n,floor(n/2)) - binomial(n+1,ceil((n-1)/2))*binomial(n,floor((n-1)/2))

from sympy import ceiling as c, binomial
def a(n):
    return binomial(n + 1, c(n/2))*binomial(n, n//2) - binomial(n + 1, c((n - 1)/2))*binomial(n, (n - 1)//2)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 02 2017

a(n) = C(n+1, ceiling(n/2))*C(n, floor(n/2)) - C(n+1, ceiling((n-1)/2))*C(n, floor((n-1)/2)). - Paul D. Hanna, Apr 16 2004
G.f.: (1/(4x^2))*((16*x^2-1)*(hypergeom([1/2, 1/2],[1],16*x^2)+2*x*(4*x-1)*hypergeom([3/2, 3/2],[2],16*x^2))-2*x+1). - Mark van Hoeij, Oct 13 2009
E.g.f (conjectured): BesselI(1,2*x)*(BesselI(0,2*x)+BesselI(1,2*x))/x. - Benjamin Phillabaum, Feb 25 2011
Conjecture: (2*n+1)*(n+3)*(n+2)*a(n) - 4*(2*n^2+4*n+3)*a(n-1) - 16*n*(2*n+3)*(n-1)*a(n-2) = 0. - R. J. Mathar, Apr 02 2017
Conjecture: (n+3)*(n+2)*a(n) - 4*(n^2+3*n+1)*a(n-1) + 16*(-n^2+n+1)*a(n-2) + 64*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Apr 02 2017
a(n) = Sum_{k=0..floor(n/2)} n!/(k!*k!*(floor(n/2)-k)!*(floor((n+1)/2)-k)!*(k+1)) (conjectured). - Roger Ford, Aug 04 2017
a(n) = A000108(floor((n+1)/2))*A000108(floor(n/2))*(2*(floor(n/2))+1). - Roger Ford, Nov 15 2019
a(n) = Product_{k=3..n} (4*floor((k-1)/2) + 2) / (floor((k+2)/2)). - Roger Ford, Apr 29 2024

A358575 Triangle read by rows where T(n,k) is the number of unlabeled n-node rooted trees with k = 0..n-1 internal (non-leaf) nodes.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 6, 1, 0, 1, 5, 14, 18, 9, 1, 0, 1, 6, 21, 39, 35, 12, 1, 0, 1, 7, 30, 72, 97, 62, 16, 1, 0, 1, 8, 40, 120, 214, 212, 103, 20, 1, 0, 1, 9, 52, 185, 416, 563, 429, 161, 25, 1

Offset: 1

Gus Wiseman, Nov 23 2022

			Triangle begins:
    1
    0    1
    0    1    1
    0    1    2    1
    0    1    3    4    1
    0    1    4    8    6    1
    0    1    5   14   18    9    1
    0    1    6   21   39   35   12    1
    0    1    7   30   72   97   62   16    1
    0    1    8   40  120  214  212  103   20    1
    0    1    9   52  185  416  563  429  161   25    1

Row sums are A000081.
Column k = n - 2 appears to be A002620.
Column k = 3 appears to be A006578.
The version for height instead of internal nodes is A034781.
Equals A055277 with rows reversed.
The ordered version is A090181 or A001263.
The central column is A185650, ordered A000891.
The left half sums to A358583, strict A358581.
The right half sums to A358584, strict A358582.
Cf. A000108, A065097, A109129, A342507, A358578, A358587, A358589.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,[_],{0,Infinity}]==k&]],{n,1,10},{k,0,n-1}]

A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.

Original entry on oeis.org

2, 6, 7, 9, 20, 22, 23, 26, 27, 29, 35, 41, 66, 76, 78, 79, 84, 86, 87, 90, 91, 93, 97, 102, 103, 106, 107, 109, 115, 117, 130, 136, 138, 139, 141, 146, 153, 163, 169, 193, 196, 197, 201, 226, 241, 262, 263, 296, 300, 302, 303, 308, 310, 311, 314, 315, 317

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The terms together with their corresponding rooted trees begin:
   2: (o)
   6: (o(o))
   7: ((oo))
   9: ((o)(o))
  20: (oo((o)))
  22: (o(((o))))
  23: (((o)(o)))
  26: (o(o(o)))
  27: ((o)(o)(o))
  29: ((o((o))))
  35: (((o))(oo))
  41: (((o(o))))
  66: (o(o)(((o))))
  76: (oo(ooo))
  78: (o(o)(o(o)))
  79: ((o(((o)))))
  84: (oo(o)(oo))
  86: (o(o(oo)))

These ordered trees are counted by A000891.
The unordered version is A358578, counted by A185650.
Height instead of leaves: counted by A358588, unordered A358576.
Height instead of internals: counted by A358590, unordered A358577.
Standard ordered tree number statistics: A358371, A358372, A358379, A358553.
A000081 counts rooted trees, ordered A000108.
A055277 counts trees by nodes and leaves, ordered A001263.
Cf. A014221, A206487, A358373, A358580, A358587, A358589.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],Count[srt[#],{},{0,Infinity}]==Count[srt[#],[_],{0,Infinity}]&]

A358371(a(n)) = A358553(a(n)).

A358588 Number of n-node ordered rooted trees of height equal to the number of internal (non-leaf) nodes.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 41, 171, 633, 2171, 7070, 22195, 67830, 203130, 598806, 1743258, 5023711, 14356226, 40737383, 114904941, 322432215, 900707165, 2506181060, 6948996085, 19207795836, 52944197508, 145567226556, 399314965956, 1093107693133, 2986640695436

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

			The a(5) = 1 and a(6) = 8 ordered trees:
  ((o)(o))  ((o)(o)o)
            ((o)(oo))
            ((o)o(o))
            ((oo)(o))
            (o(o)(o))
            (((o))(o))
            (((o)(o)))
            ((o)((o)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For leaves instead of height we have A000891, unordered A185650 aerated.
The unordered version is A358587, ranked by A358576.
For leaves instead of internal nodes we have A358590, unordered A358589.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
Cf. A065097, A342507, A358552, A358371, A358578, A358579, A358586, A358591.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,[_],{0,Infinity}]==Depth[#]-1&]],{n,1,10}]

\\ Needs R(n,f) defined in A358590.
seq(n) = {Vec(R(n, (h,p)->polcoef(subst(p, x, x/y), -h, y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Conjectures from Chai Wah Wu, Apr 14 2024: (Start)
a(n) = 9*a(n-1) - 32*a(n-2) + 58*a(n-3) - 58*a(n-4) + 32*a(n-5) - 9*a(n-6) + a(n-7) for n > 7.
G.f.: x^5*(-x^2 + x - 1)/((x - 1)^3*(x^2 - 3*x + 1)^2). (End)

Terms a(16) and beyond from Andrew Howroyd, Jan 01 2023

A000356 Number of rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).

Original entry on oeis.org

1, 5, 35, 294, 2772, 28314, 306735, 3476330, 40831076, 493684828, 6114096716, 77266057400, 993420738000, 12964140630900, 171393565105575, 2291968851019650, 30961684478686500, 422056646314726500

Offset: 1

N. J. A. Sloane, Simon Plouffe

a(2n-1) is also the sum of the numbers of standard Young tableaux of size 2n+1 and of shapes (k+3,k+2,2^{n-2-k}), 0 <= k <= n-2. - Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010

Amitai Regev, Preprint. [From Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..800
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
W. T. Tutte, On the enumeration of four-colored maps, SIAM J. Appl. Math., 17 (1969), 454-460.

Cf. A000264, A000309.
Equals A005568/2.
Fourth row of array A102539.
Column of array A073165.
Image of A001700 under the "little Hankel" transform (see A056220 for definition). - John W. Layman, Aug 22 2000
Cf. A000891.

A000356 := proc(n)
    binomial(2*n,n)*binomial(2*n+1,n+1)/(n+1)/(n+2) ;
end proc:

CoefficientList[ Series[1 + (HypergeometricPFQ[{1, 3/2, 5/2}, {3, 4}, 16 x] - 1), {x, 0, 17}], x]
Table[(2*n)!*(2*n+2)!/(2*n!*(n+1)!^2*(n+2)!),{n,30}] (* Vincenzo Librandi, Mar 25 2012 *)

G.f.: (with offset 0) 3F2( [1, 3/2, 5/2], [3, 4], 16*x) = (1 - 2*x - 2F1( [-1/2, 1/2], [2], 16*x) ) / (4*x^2). - Olivier Gérard, Feb 16 2011
a(n)*(n+2) = A000891(n). - Gary W. Adamson, Apr 08 2011
D-finite with recurrence (n+2)*(n+1)*a(n)-4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 03 2013
From Ilya Gutkovskiy, Feb 01 2017: (Start)
E.g.f.: (1/2)*(2F2(1/2,3/2; 2,3; 16*x) - 1).
a(n) ~ 2^(4*n+1)/(Pi*n^3). (End)
From Peter Bala, Feb 22 2023: (Start)
a(n) = Product_{1 <= i <= j <= n-1} (i + j + 3)/(i + j - 1).
a(n) = (2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 3)/(i + j) for n >= 1.
 Cf. A003645. (End)

Better definition from Michael Albert, Oct 24 2008

A103905 Square array T(n,k) read by antidiagonals: number of tilings of an hexagon.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 20, 20, 4, 1, 70, 175, 50, 5, 1, 252, 1764, 980, 105, 6, 1, 924, 19404, 24696, 4116, 196, 7, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1, 48620, 34763300

Offset: 1

Ralf Stephan, Feb 22 2005

As a square array, T(n,k) = number of all k-watermelons without a wall of length n. - Steven Finch, Mar 30 2008

			Array begins:
  1,   2,     3,      4,        5,         6, ...
  1,   6,    20,     50,      105,       196, ...
  1,  20,   175,    980,     4116,     14112, ...
  1,  70,  1764,  24696,   232848,   1646568, ...
  1, 252, 19404, 731808, 16818516, 267227532, ...
  ...

Peter J. Forrester and Alex Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.
Anthony J. Guttmann, Aleksandr L. Owczarek and Xavier G. Viennot, Vicious walkers and Young tableaux. I. Without walls, J. Phys. A 31 (1998) 8123-8135.
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
Christian Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507 [math.CO], 2005.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 28.
Percy A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.

Rows include A002415, A047819, A047835, A047831.
Columns include A000984 and A000891.
Main diagonal is A008793.
Cf. A120258, A133112.

t[n_, k_] := Product[j!*(j + 2*n)!/(j + n)!^2, {j, 0, k - 1}]; Join[{1}, Flatten[ Table[ t[n - k , k], {n, 1, 10}, {k, 1, n}]]] (* Jean-François Alcover, May 16 2012, from 2nd formula *)

T(n, k) = [V(2n+k-1)V(k-1)V(n-1)^2]/[V(2n-1)V(n+k-1)^2], with V(n) the superfactorial numbers (A000178).
T(n, k) = Prod[j=0..k-1, j!(j+2n)!/(j+n)!^2 ].
T(n, k) = Prod[h=1..n, Prod[i=1..k, Prod[j=1..n, (h+i+j-1)/(h+i+j-2) ]]].
T(n, k) = Prod[i=1..k, Prod[j=n+1..2n+1, i+j]/Prod[j=0..n, i+j]]; - Paul Barry, Jun 13 2006
Conjectural formula as a sum of squares of Vandermonde determinants: T(n,k) = 1/((1!*2! ... *(n-1)!)^2*n!)* sum {1 <= x_1, ..., x_n <= k} (det V(x_1, ..., x_n))^2, where V(x_1, ..., x_n) is the Vandermonde matrix of order n. Compare with A133112. - Peter Bala, Sep 18 2007
For k >= 1, T(n,k)=det(binomial(2*n,n+i-j))1<=i,j<=k [Krattenhaller, Theorem 4].
Let H(n) = product {k = 1..n-1} k!. Then for a,b,c nonnegative integers (H(a)*H(b)*H(c)*H(a+b+c))/(H(a+b)*H(b+c)*H(c+a)) is an integer [MacMahon, Section 4.29 with x -> 1]. Setting a = b = n and c = k gives the entries for this table. - Peter Bala, Dec 22 2011

A123610 Triangle read by rows, where T(n,k) = (1/n)Sum_{d|(n,k)} phi(d) binomial(n/d,k/d)^2 for n >= k > 0, with T(n,0) = 1 for n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 20, 20, 5, 1, 1, 6, 39, 68, 39, 6, 1, 1, 7, 63, 175, 175, 63, 7, 1, 1, 8, 100, 392, 618, 392, 100, 8, 1, 1, 9, 144, 786, 1764, 1764, 786, 144, 9, 1, 1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1, 1, 11, 275, 2475, 9900

Offset: 0

Paul D. Hanna, Oct 03 2006

A variant of the triangle A047996 of circular binomial coefficients.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  4,  10,    4,    1;
  1,  5,  20,   20,    5,    1;
  1,  6,  39,   68,   39,    6,    1;
  1,  7,  63,  175,  175,   63,    7,    1;
  1,  8, 100,  392,  618,  392,  100,    8,   1;
  1,  9, 144,  786, 1764, 1764,  786,  144,   9,  1;
  1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1;
  ...
Example of column g.f.s are:
column 1: 1/(1 - x)^2;
column 2: Ser([1, 1, 3, 1]) / ((1 - x)^2*(1 - x^2)^2) = g.f. of A005997;
column 3: Ser([1, 2, 11, 26, 30, 26, 17, 6, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 -x^3)^2);
column 4: Ser([1, 3, 28, 94, 240, 440, 679, 839, 887, 757, 550, 314, 148, 48, 11, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2*(1 - x^4)^2);
where Ser() denotes a polynomial in x with the given coefficients, as in Ser([1, 1, 3, 1]) = (1 + x + 3*x^2 + x^3).

Paul D. Hanna, Rows n = 0..45, flattened.
Petros Hadjicostas, Proofs of some formulae for g.f.'s of this sequence.

Cf. Columns: A005997, A123613, A123614, A123615, A123616.
Cf. A123611 (row sums), A123612 (antidiagonal sums), A123617 (central terms).
Cf. A123618, A123619, A047996 (variant), A128545.

T[, 0] = 1; T[n, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]* Binomial[n/#, k/#]^2, 0]&]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

{T(n,k)=if(k==0,1,(1/n)*sumdiv(n,d,if(gcd(k,d)==d, eulerphi(d)*binomial(n/d,k/d)^2,0)))}

T(2*n+1, n) = (2*n + 1)*A000108(n)^2 = (2*n + 1)*((2*n)!/(n!(n+1)!))^2 = A000891(n) for n >= 0.
Row sums are 2*A047996(2*n,n) = 2*A003239(n) for n > 0.
Row sums equal the row sums of triangle A128545.
For n >= 1, the g.f. of column n has the form: P_n(x)/(Product_{m=1..n} (1 - x^m)^2), where P_n(x) is a polynomial with n^2 coefficients such that the sum of the coefficients is P_n(1) = (2*n - 1)!.
From Petros Hadjicostas, Oct 24 2017: (Start)
Proofs of the following formulae can be found in the links.
G.f.: Sum_{n>=1, k>=0} T(n,k)*x^n*y^k = -Sum_{s>=1} (phi(s)/s)*log(f(x^s,y^s)), where phi(s) is Euler's totient function at s, f(x,y) = (sqrt(g(x,y)) + 1 -(1 + y)*x)/2, and g(x,y) = 1 - 2*(1 + y)*x + (1 - y)^2*x^2. (Term T(0,0) is not used in this g.f.)
Row g.f.: Sum_{k>=0} T(n,k)*y^k = (1/n)*Sum_{d|n} phi(d)*R(n/d, y^d), where R(m, y) = [z^m] (1 + (1 + y)*z + y*z^2)^m. (End)

A145600 a(n) is the number of walks from (0,0) to (0,1) that remain in the upper half-plane y >= 0 using (2*n - 1) unit steps either up (U), down (D), left (L) or right (R).

Original entry on oeis.org

1, 8, 75, 784, 8820, 104544, 1288287, 16359200, 212751396, 2821056160, 38013731756, 519227905728, 7174705330000, 100136810390400, 1409850293610375, 20002637245262400, 285732116760449700

Offset: 1

Peter Bala, Oct 14 2008

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145601, A145602 and A145603. This sequence is the central column taken from triangle A145596, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 1.

			a(2) = 8: the 8 walks from (0,0) to (0,1) of three steps are
UDU, UUD, URL, ULR, RLU, LRU, RUL and LUR.

M. Dukes and Y. Le Borgne, Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013, Pages 816-842. - From N. J. A. Sloane, Feb 21 2013

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

Cf. A000891, A145596, A145601, A145602, A145603.

a(n) := 1/n*binomial(2*n,n+1)*binomial(2*n,n-1);
seq(a(n),n = 1..19);

a(n) = 1/n*binomial(2*n,n+1)*binomial(2*n,n-1).
a(n) = A135389(n-1)/(n+1). - R. J. Mathar, Jul 14 2013
D-finite with recurrence (n+1)^2*a(n) -4*n*(5*n-1)*a(n-1) +16*(2*n-3)^2*a(n-2)=0. - R. J. Mathar, Jul 14 2013

A358584 Number of rooted trees with n nodes, at most half of which are leaves.

Original entry on oeis.org

0, 1, 1, 3, 5, 15, 28, 87, 176, 550, 1179, 3688, 8269, 25804, 59832, 186190, 443407, 1375388, 3346702, 10348509, 25632265, 79020511, 198670299, 610740694, 1555187172, 4768244803, 12276230777, 37546795678, 97601239282, 297831479850, 780790439063, 2377538260547

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(2) = 1 through a(6) = 15 trees:
  (o)  ((o))  ((oo))   (((oo)))   (((ooo)))
              (o(o))   ((o)(o))   ((o)(oo))
              (((o)))  ((o(o)))   ((o(oo)))
                       (o((o)))   ((oo(o)))
                       ((((o))))  (o((oo)))
                                  (o(o)(o))
                                  (o(o(o)))
                                  (oo((o)))
                                  ((((oo))))
                                  (((o)(o)))
                                  (((o(o))))
                                  ((o)((o)))
                                  ((o((o))))
                                  (o(((o))))
                                  (((((o)))))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For equality we have A185650 aerated, ranked by A358578.
The complement is A358581.
The strict case is A358582.
The opposite version is A358583.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
A358589 counts square trees, ranked by A358577, ordered A358590.
Cf. A000891, A034781, A065097, A109129, A342507, A358579, A358580, A358585, A358586, A358591.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}]<=Count[#,[_],{0,Infinity}]&]],{n,0,10}]

R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + O(x*x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
seq(n) = {my(A=R(n)); vector(n, n, vecsum(Vecrev(A[n]/y)[1..n\2]))} \\ Andrew Howroyd, Dec 30 2022

A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=1..floor(n/2)} A055277(n, k). - Andrew Howroyd, Dec 30 2022

Terms a(19) and beyond from Andrew Howroyd, Dec 30 2022

cp's OEIS Frontend

A358581 Number of rooted trees with n nodes, most of which are leaves.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A005558 a(n) is the number of n-step walks on square lattice such that 0 <= y <= x at each step.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A358575 Triangle read by rows where T(n,k) is the number of unlabeled n-node rooted trees with k = 0..n-1 internal (non-leaf) nodes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A358579 Numbers k such that the k-th standard ordered rooted tree has the same number of leaves as internal (non-leaf) nodes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A358588 Number of n-node ordered rooted trees of height equal to the number of internal (non-leaf) nodes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A000356 Number of rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A103905 Square array T(n,k) read by antidiagonals: number of tilings of an hexagon.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A123610 Triangle read by rows, where T(n,k) = (1/n)Sum_{d|(n,k)} phi(d) binomial(n/d,k/d)^2 for n >= k > 0, with T(n,0) = 1 for n >= 0.