A090181 -id:A090181 - cp's OEIS frontend

A358591 Number of 2n-node rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

0, 0, 2, 17, 94, 464, 2162, 9743, 42962, 186584, 801316, 3412034, 14430740, 60700548, 254180426, 1060361147, 4409342954, 18285098288, 75645143516, 312286595342, 1286827096964, 5293833371408, 21745951533236, 89208948855542, 365523293690804, 1496048600896784

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(3) = 2 and a(4) = 17 trees:
  ((o)(oo))  (((o))(ooo))
  (o(o)(o))  (((o)(ooo)))
             (((oo))(oo))
             (((oo)(oo)))
             ((o)((ooo)))
             ((o)(o(oo)))
             ((o)(oo(o)))
             ((o(o)(oo)))
             ((oo)(o(o)))
             ((oo(o)(o)))
             (o((o))(oo))
             (o((o)(oo)))
             (o(o)((oo)))
             (o(o)(o(o)))
             (o(o(o)(o)))
             (oo((o)(o)))
             (oo(o)((o)))

Andrew Howroyd, Table of n, a(n) for n = 1..100

For leaves = internals we have A185650 aerated, ranked by A358578.
For height = internals we have A358587, ranked by A358576, ordered A358588.
For height = leaves we have A358589, ranked by A358577, ordered A358590.
These trees are ranked by A358592.
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.
Cf. A065097, A109082, A109129, A342507, A358552, A358581-A358586.

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}]==Depth[#]-1&]],{n,2,15,2}]

\\ Needs R(n,f) defined in A358589.
seq(n) = {Vecrev(R(2*n, (h,p)->if(h<=n, x^h*polcoef(polcoef(p, 2*h, x), h, y))), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023

A358582 Number of rooted trees with n nodes, most of which are not leaves.

Original entry on oeis.org

0, 0, 1, 1, 5, 7, 28, 48, 176, 336, 1179, 2420, 8269, 17855, 59832, 134289, 443407, 1025685, 3346702, 7933161, 25632265, 62000170, 198670299, 488801159, 1555187172, 3882403641, 12276230777, 31034921462, 97601239282, 249471619165, 780790439063, 2015194486878

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(3) = 1 through a(6) = 7 trees:
  ((o))  (((o)))  (((oo)))   ((((oo))))
                  ((o)(o))   (((o)(o)))
                  ((o(o)))   (((o(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 opposite version is A358581, non-strict A358583.
The non-strict version is A358584.
The ordered version is A358585, odd-indexed terms A065097.
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, 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}][_],{0,Infinity}]&]],{n,0,10}]

\\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, vecsum(Vecrev(A[n]/y)[1..(n-1)\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-1)/2)} A055277(n, k). - Andrew Howroyd, Dec 30 2022

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

A358585 Number of ordered rooted trees with n nodes, most of which are leaves.

Original entry on oeis.org

1, 0, 1, 1, 7, 11, 66, 127, 715, 1549, 8398, 19691, 104006, 258194, 1337220, 3467115, 17678835, 47440745, 238819350, 659060677, 3282060210, 9271024542, 45741281820, 131788178171, 644952073662, 1890110798926, 9183676536076, 27316119923002, 131873975875180, 397407983278484

Offset: 1

Gus Wiseman, Nov 24 2022

nonn

			The a(1) = 1 through a(6) = 11 ordered trees:
  o  .  (oo)  (ooo)  (oooo)   (ooooo)
                     ((o)oo)  ((o)ooo)
                     ((oo)o)  ((oo)oo)
                     ((ooo))  ((ooo)o)
                     (o(o)o)  ((oooo))
                     (o(oo))  (o(o)oo)
                     (oo(o))  (o(oo)o)
                              (o(ooo))
                              (oo(o)o)
                              (oo(oo))
                              (ooo(o))

Andrew Howroyd, Table of n, a(n) for n = 1..1000

For equality we have A000891, unordered A185650.
Odd-indexed terms are A065097.
The unordered version is A358581.
The opposite is the same, unordered A358582.
The non-strict version is A358586, unordered A358583.
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.
A358590 counts square ordered trees, unordered A358589 (ranked by A358577).
Cf. A109129, A342507, A358371, A358579, A358580, A358584, A358588, A358590.

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

a(n) = if(n==1, 1, n--; (binomial(2*n,n)/(n+1) - if(n%2, binomial(n, (n-1)/2)^2 / n))/2) \\ Andrew Howroyd, Jan 13 2024

From Andrew Howroyd, Jan 13 2024: (Start)
a(n) = Sum_{k=1..floor((n-1)/2)} A001263(n-1, k) for n >= 2.
a(2*n) = (A000108(2*n-1) - A000891(n-1))/2 for n >= 1;
a(2*n+1) = A000108(2*n)/2 for n >= 1. (End)

a(16) onwards from Andrew Howroyd, Jan 13 2024

A131198 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,1,0,1,0,1,0,...] DELTA [0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0

Offset: 0

Philippe Deléham, Oct 20 2007

Mirror image of triangle A090181, another version of triangle of Narayana (A001263).

Equals A133336*A130595 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007

			Triangle begins:
  1;
  1,  0;
  1,  1,   0;
  1,  3,   1,   0;
  1,  6,   6,   1,   0;
  1, 10,  20,  10,   1,   0;
  1, 15,  50,  50,  15,   1,  0;
  1, 21, 105, 175, 105,  21,  1, 0;
  1, 28, 196, 490, 490, 196, 28, 1, 0; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), Article 09.7.6.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011), Article 11.4.5.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
FindStat - Combinatorial Statistic Finder, The number of peaks of a Dyck path., The number of double rises of a Dyck path., The number of valleys of a Dyck path., The number of left oriented leafs except the first one of a binary tree., The number of left tunnels of a Dyck path.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Cf. A000217, A002415, A006542, A006857.

[[n le 0 select 1 else (n-k)*Binomial(n,k)^2/(n*(k+1)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018

T := (n,k) -> `if`(n=0, 0^n, binomial(n,k)^2*(n-k)/(n*(k+1)));
seq(print(seq(T(n,k), k=0..n)), n=0..5); # Peter Luschny, Jun 08 2014
R := n -> simplify(hypergeom([1 - n, -n], [2], x)):
Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
seq(print(Trow(n)), n = 0..9); # Peter Luschny, Apr 26 2022

Table[If[n == 0, 1, (n-k)*Binomial[n,k]^2/(n*(k+1))], {n,0,10}, {k,0,n}] //Flatten (* G. C. Greubel, Feb 06 2018 *)

for(n=0,10, for(k=0,n, print1(if(n==0,1, (n-k)*binomial(n,k)^2/(n* (k+1))), ", "))) \\ G. C. Greubel, Feb 06 2018

Sum_{k=0..n} T(n,k)*x^k = A000012(n), A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Oct 23 2007
Sum_{k=0..floor(n/2)} T(n-k,k) = A004148(n). - Philippe Deléham, Nov 06 2007
T(2*n,n) = A125558(n). - Philippe Deléham, Nov 16 2011
T(n, k) = [x^k] hypergeom([1 - n, -n], [2], x). - Peter Luschny, Apr 26 2022

A358583 Number of rooted trees with n nodes, at least half of which are leaves.

Original entry on oeis.org

1, 1, 1, 3, 4, 13, 20, 67, 110, 383, 663, 2346, 4217, 15118, 27979, 101092, 191440, 695474, 1341974, 4893067, 9589567, 35055011, 69612556, 254923825, 511987473, 1877232869, 3807503552, 13972144807, 28585315026, 104955228432, 216381073935, 794739865822

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(1) = 1 through a(6) = 13 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)
                ((oo))  ((ooo))  ((oooo))
                (o(o))  (o(oo))  (o(ooo))
                        (oo(o))  (oo(oo))
                                 (ooo(o))
                                 (((ooo)))
                                 ((o)(oo))
                                 ((o(oo)))
                                 ((oo(o)))
                                 (o((oo)))
                                 (o(o)(o))
                                 (o(o(o)))
                                 (oo((o)))

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

For equality we have A185650 aerated, ranked by A358578.
The strict case is A358581.
The opposite version is A358584, strict A358582.
The ordered version is A358586, strict A358585.
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 rooted trees, ranked by A358577, ordered A358590.
Cf. A000891, A034781, A065097, A109129, A342507, A358579, A358580, A358586.

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,1,10}]

\\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[(n-1)\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-1)/2)+1..n} A055277(n, k). - Andrew Howroyd, Dec 31 2022

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

A154825 Reversion of x(1-2x)/(1-3*x).

Original entry on oeis.org

1, -1, -1, 1, 5, 3, -21, -51, 41, 391, 407, -1927, -6227, 2507, 49347, 71109, -236079, -966129, 9519, 7408497, 13685205, -32079981, -167077221, -60639939, 1209248505, 2761755543, -4457338681, -30629783831, -22124857219, 206064020315, 572040039283, -590258340811

Offset: 0

Paul Barry, Jan 15 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A060693, A086810, A088617, A090181, A091593, A131198, A133336.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1+3*x-Sqrt(1-2*x+9*x^2))/(4*x) )); // G. C. Greubel, May 24 2022

A154825_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := -a[w-1]+2*add(a[j]*a[w-j-1],j=1..w-1)od;
convert(a, list) end: A154825_list(28); # Peter Luschny, May 19 2011

CoefficientList[Series[(1+3*x-Sqrt[1-2*x+9*x^2])/(4*x), {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

[sum(binomial(n+k,n-k)*catalan_number(k)*2^k*(-3)^(n-k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, May 24 2022

G.f.: (1+3*x-sqrt(1-2*x+9*x^2))/(4*x). - corrected by Vaclav Kotesovec, Feb 08 2014
G.f.: 1/(1+x/(1-2x/(1+x/(1-2x/(1+x/(1-2x/(1+.... (continued fraction).
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*A000108(k)*2^k*(-3)^(n-k).
From Philippe Deléham, Jan 17 2009: (Start)
a(n) = Sum_{k=0..n} A131198(n,k)*(-1)^(n-k)*2^k.
a(n) = Sum_{k=0..n} A090181(n,k)*(-1)^k*2^(n-k).
a(n) = Sum_{k=0..n} A060693(n,k)*2^(n-k)*(-3)^k.
a(n) = Sum_{k=0..n} A088617(n,k)*2^k*(-3)^(n-k).
a(n) = Sum_{k=0..n} A086810(n,k)*(-1)^k*3^(n-k).
a(n) = Sum_{k=0..n} A133336(n,k)*3^k*(-1)^(n-k). (End)
D-finite with recurrence (n+1)*a(n) = (2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - R. J. Mathar, Nov 15 2012
a(n) = (-3)^n*Hypergeometric2F1([-n, n+1], [2]; 2/3). - G. C. Greubel, May 24 2022

A359363 Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 10, 1, 0, 1, 20, 50, 20, 1, 0, 1, 35, 175, 175, 35, 1, 0, 1, 56, 490, 980, 490, 56, 1, 0, 1, 84, 1176, 4116, 4116, 1176, 84, 1, 0, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1, 0, 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1

Offset: 0

Peter Luschny, Dec 28 2022

This triangle is a member of a family of Pascal-like triangles. Let T(n, k, m) = sf(m)*F(n - 1) / (F(k - 1)*F(n - k)) if k > 0 and otherwise k^n, where F(n) = Product_{j=0..m} (n + j)! and sf(m) are the superfactorials A000178. The case m = 2 gives this triangle, some other cases are given in the crossreferences. See also A342889 for a related representation of generalized binomial coefficients.

			Triangle T(n, k) starts:
[0] 1
[1] 0, 1
[2] 0, 1,   1
[3] 0, 1,   4,    1
[4] 0, 1,  10,   10,     1
[5] 0, 1,  20,   50,    20,     1
[6] 0, 1,  35,  175,   175,    35,     1
[7] 0, 1,  56,  490,   980,   490,    56,    1
[8] 0, 1,  84, 1176,  4116,  4116,  1176,   84,   1
[9] 0, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1
.
Let p = (p1, p2,..., pn) denote a permutation of {1, 2,..., n}. The pair (p(i), p(i+1)) is a 'rise' if p(i) < p(i+1). Additionally a conventional rise is counted at the beginning of p.
T(n, k) is the number of Baxter permutations of {1,2,...,n} with k rises. For example for n = 4, [T(n, k) for k = 0..n] = [0, 1, 10, 10, 1]. The permutations, with preceding number of rises, are:
.
1 [4, 3, 2, 1],  3 [2, 3, 4, 1],  2 [3, 4, 2, 1],  3 [2, 3, 1, 4],
2 [3, 2, 4, 1],  3 [2, 1, 3, 4],  2 [3, 2, 1, 4],  3 [1, 3, 4, 2],
2 [2, 4, 3, 1],  3 [1, 3, 2, 4],  2 [4, 2, 3, 1],  3 [3, 4, 1, 2],
2 [2, 1, 4, 3],  3 [3, 1, 2, 4],  2 [4, 2, 1, 3],  3 [1, 2, 4, 3],
2 [1, 4, 3, 2],  3 [1, 4, 2, 3],  2 [4, 1, 3, 2],  3 [4, 1, 2, 3],
2 [4, 3, 1, 2],  4 [1, 2, 3, 4].

Special cases of the general formula: A097805 (m = 0), (0,1)-Pascal triangle; A090181 (m = 1), triangle of Narayana; this triangle (m = 2); A056940 (m = 3), with 1,0,0...; A056941 (m = 4), with 1,0,0...; A142465 (m = 5), with 1,0,0....
Variant: A056939. Diagonals: A000292, A006542, A047819.
Cf. A000178, A001181, A046996, A217800, A342889.

p := (n, x) -> ifelse(n = 0, 1, x*hypergeom([-1-n, -n, 1-n], [2, 3], -x)):
seq(seq(coeff(simplify(p(n, x)), x, k), k = 0..n), n = 0..10);
# Alternative:
T := proc(n, k) local F; F := n -> n!*(n+1)!*(n+2)!;
ifelse(k = 0, k^n, 2*F(n-1)/(F(k-1)*F(n-k))) end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

C=binomial;
T(n, k) = if(n==0 && k==0, 1, ( C(n+1,k-1) * C(n+1,k) * C(n+1,k+1) ) / ( C(n+1,1) * C(n+1,2) ) );
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print());
\\ Joerg Arndt, Jan 04 2024

from functools import cache
from math import factorial
@cache
def A359363Row(n: int) -> list[int]:
    @cache
    def F(n: int): return factorial(n) ** 3 * ((n+1) * (n+1) * (n+2))
    if n == 0: return [1]
    return [0] + [(2*F(n-1))//(F(k-1) * F(n-k)) for k in range(1, n+1)]
for n in range(0, 10): print(A359363Row(n))
# Peter Luschny, Jan 04 2024

def A359363(n):
    if n == 0: return SR(1)
    h = x*hypergeometric([-1 - n, -n, 1 - n], [2, 3], -x)
    return h.series(x, n + 1).polynomial(SR)
for n in range(10): print(A359363(n).list())
def PolyA359363(n, t): return Integer(A359363(n)(x=t).n())
# Peter Luschny, Jan 04 2024

T(n, k) = [x^k] p(n, x).
T(n, k) = 2*F(n-1)/(F(k-1)*F(n-k)) for k > 0 where F(n) = n!*(n+1)!*(n+2)!.
p(n, 1) = A001181(n), i.e. the Baxter numbers are the values of the Baxter polynomials at x = 1.
(-1)^(n + 1)*p(2*n + 1, -1) = A217800(n) .

A155926 G.f. satisfies: A(x) = B(xA(x)) where A(x) = Sum_{n>=0} a(n)x^n/[n!(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!(n+1)!/2^n].

Original entry on oeis.org

1, 1, 4, 37, 621, 16526, 640207, 34039027, 2379382609, 211619306134, 23337543447296, 3125553148981176, 499716551101393705, 94016487294245251308, 20561796731966531616954, 5172827581575899147920471

Offset: 0

Paul D. Hanna, Jan 30 2009

nonn

			G.f.: A(x) = 1 + x + 4*x^2/3 + 37*x^3/18 + 621*x^4/180 + 16526*x^5/2700 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
B(x) = 1 + x + 1/3*x^2 + 1/18*x^3 + 1/180*x^4 +...+ x^n/[n!*(n+1)!/2^n] +... where
A(x) = B(x*A(x)) and B(x) = A(x/B(x)) ;
1/B(x) = 1 - x + 2*x^2/3 - 7*x^3/18 + 39*x^4/180 - 321*x^5/2700 +...+ (-1)^n*A103365(n)*x^n/[n!*(n+1)!/2^n] +...
Also, A(x) = C(x*A(x)^2) where:
C(x) = 1 + x - 2*x^2/3 + 19*x^3/18 - 379*x^4/180 + 12726*x^5/2700 +...+ A155927(n)*x^(n+1)/[n!*(n+1)!/2^n] +...
A(x)^2 = 1 + 2*x + 11*x^2/3 + 122*x^3/18 + 2302*x^4/180 + 66482*x^5/2700 +...

Cf. A105558, A105556, A001263, A103365, A006472, A090181, A155927, A155928 (A^2).

{a(n)=local(F=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(serreverse(x/F)/x,n)*n!*(n+1)!/2^n}

{a(n)=local(N=matrix(n+1, n+1, m, j, if(m>=j, binomial(m-1, j-1)*binomial(m, j-1)/j))); sum(j=0, n, (N^n)[n+1, j+1])/(n+1)}

a(n) = A105558(n)/(n+1) = A105556(2n,n)/(n+1) = [N^(n+1)](n+1,1)/(n+1) for n>=0, where N^(n+1) is the (n+1)-th matrix power of the Narayana triangle N=A001263.
G.f.: A(x) = Series_Reversion[x/B(x)]/x where B(x) = A(x/B(x)) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n].
G.f. satisfies: A(x) = C(x*A(x)^2) and C(x) = A(x/C(x)^2) where C(x) is the g.f. of A155927.

A143749 Series reversion of x * (1 - x) / (1 + 9*x).

Original entry on oeis.org

0, 1, 10, 110, 1310, 16610, 221010, 3051510, 43357110, 630098810, 9324499610, 140046944510, 2129440330510, 32716182966610, 507115641523810, 7920881045935110, 124548017695545510, 1969917348711212010

Offset: 0

Paul Barry, Aug 30 2008

Hankel transform of a(n) is A143750. Hankel transform of a(n+1) is 10^C(n+1,2).

			G.f. = x + 10*x^2 + 110*x^3 + 1310*x^4 + 16610*x^5 + 221010*x^6 + 3051510*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A090181, A143750.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-9*x-Sqrt(81*x^2-22*x+1))/(2*x))); // G. C. Greubel, Sep 16 2018

CoefficientList[Series[(1-9*x-Sqrt[1-22*x+81*x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = 9 * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

G.f.: (1-9*x-sqrt(1-22*x+81*x^2))/2.
a(n) = Sum_{k=0..n-1} C(n+k-1,2*k)*A000108(k)*9^(n-k-1).
a(n+1) = Sum_{k=0..n} C(2*n-k,k)*A000108(n-k)*9^k.
a(n+1) = 0^n + (1/(n+0^n)) * Sum_{k=0..n} C(n,k)*C(n,k-1)*10^k.
a(n+1) = Sum_{k, 0<=k<=n} 10^k*A090181(n,k). - Philippe Deléham, Oct 14 2008
a(n) = 9 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
Recurrence: n*a(n) = 11*(2*n-3)*a(n-1) - 81*(n-3)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ sqrt(11*sqrt(10)-20)*(11+2*sqrt(10))^n/(18*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012
0 = a(n)*(6561*a(n+1) - 4455*a(n+2) + 324*a(n+3)) + a(n+1)*(891*a(n+1) + 322*a(n+2) - 55*a(n+3)) + a(n+2)*(11*a(n+2) + a(n+3)) if n>0. - Michael Somos, Jan 23 2014
G.f.: x/(1 - 9*x - x/(1 - 9*x - x/(1 - 9*x - x/(1 - 9*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 07 2018

A152601 a(n) = Sum_{k=0..n} C(n+k,2k)A000108(k)3^k*2^(n-k).

Original entry on oeis.org

1, 5, 40, 395, 4360, 51530, 637840, 8163095, 107140360, 1434252230, 19507077040, 268796321870, 3744480010960, 52647783144980, 746145741252640, 10648007952942095, 152877753577617160, 2206713692628578030

Offset: 0

Paul Barry, Dec 09 2008

Hankel transform is 15^C(n+1,2).

Cf. A103211, A103210.
Cf. A088617, A060693, A090181, A131198, A133336, A086810
Cf. A152600.

a(n) = A152600(n+1)/2.
a(n) = Sum_{k=0..n} A088617(n,k)*3^k*2^(n-k) = Sum_{k=0..n} A060693(n,k)*2^k*3^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A090181(n,k)*5^k*3^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A131198(n,k)*3^k*5^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A133336(n,k)*(-2)^k*5^(n-k) = Sum_{k=0..n} A086810(n,k)*5^k*(-2)^(n-k). - Philippe Deléham, Dec 10 2008
G.f.: 1/(1-5x/(1-3x/(1-5x/(1-3x/(1-5x/(1-3x/(1-5x/(1-... (continued fraction). - Philippe Deléham, Nov 28 2011
Conjecture: (n+1)*a(n) +8*(-2*n+1)*a(n-1) +4*(n-2)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
G.f.: 1/G(x), with G(x) = 1-2*x-(3*x)/G(x) (continued fraction). - Nikolaos Pantelidis, Jan 09 2023

cp's OEIS Frontend

A358591 Number of 2n-node rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

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A358582 Number of rooted trees with n nodes, most of which are not leaves.

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A358585 Number of ordered rooted trees with n nodes, most of which are leaves.

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A131198 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,1,0,1,0,1,0,...] DELTA [0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938.

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A358583 Number of rooted trees with n nodes, at least half of which are leaves.

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A154825 Reversion of x*(1-2*x)/(1-3*x).

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A359363 Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 1.

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A154825 Reversion of x(1-2x)/(1-3*x).

A155926 G.f. satisfies: A(x) = B(xA(x)) where A(x) = Sum_{n>=0} a(n)x^n/[n!(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!(n+1)!/2^n].

A152601 a(n) = Sum_{k=0..n} C(n+k,2k)A000108(k)3^k*2^(n-k).