A065097 -id:A065097 - cp's OEIS frontend

A265102 a(n) = binomial(8n + 6, 4n + 1)/(8*n + 6).

1, 143, 22610, 3991995, 757398510, 150946230006, 31170212479588, 6611198199648595, 1431806849011462742, 315319074704135127010, 70398290295706497441660, 15897587681946817926283230, 3624898901185998294920196300, 833406923656808938891174678092

Offset: 0

Peter Bala, Dec 02 2015

Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/4).

Number of maximal faces of the rational associahedron Ass(4*n + 1, 4*n + 5). Number of lattice paths from (0, 0) to (4*n + 5, 4*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (4*n + 1)/(4*n + 5)*x. See Armstrong et al.

Seiichi Manyama, Table of n, a(n) for n = 0..416
D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Row 4 of A306444.

Cf. A000108, A065097 (Cat(n + 1/2)), A265101 (Cat(n + 1/3)), A265103 (Cat(n + 1/5)).

[Binomial(8*n+6, 4*n+1)/(8*n+6): n in [0..20]]; // G. C. Greubel, Feb 16 2019

seq(binomial(8*n + 6, 4*n + 1)/(8*n + 6), n = 0..14);

Table[Binomial[8n+6, 4n+1]/(8n+6), {n, 0, 20}] (* Vincenzo Librandi, Dec 09 2015 *)

a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6); \\ Altug Alkan, Dec 07 2015

[binomial(8*n+6, 4*n+1)/(8*n+6) for n in (0..20)] # G. C. Greubel, Feb 16 2019

a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6).
(n + 1)*(2*n - 1)*(4*n + 3)*(4*n + 5)*a(n) = 2*(8*n + 1)*(8*n - 1)*(8*n + 3)*(8*n + 5)*a(n-1) with a(0) = 1.
From Ilya Gutkovskiy, Feb 28 2017: (Start)
O.g.f.: (4F3(-1/8,1/8,3/8,5/8; -1/2,3/4,5/4; 256*x) - 1)/(2*x).
E.g.f.: 4F4(7/8,9/8,11/8,13/8; 1/2,7/4,2,9/4; 256*x).
a(n) ~ 4^(4*n+1)/(sqrt(Pi)*n^(3/2)). (End)

A265103 a(n) = binomial(10n + 7, 5n + 1)/(10*n + 7).

Original entry on oeis.org

1, 728, 482885, 347993910, 267058714626, 214401560777712, 177957899774070416, 151516957974714281810, 131614194900668669130060, 116186564091895720987588128, 103938666796148178180041038716, 94020887900502277905668153549928, 85855382816448334044679630209920925

Offset: 0

Peter Bala, Dec 02 2015

Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/5).

Number of maximal faces of the rational associahedron Ass(5*n + 1, 5*n + 6). Number of lattice paths from (0, 0) to (5*n + 6, 5*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (5*n + 1)/(5*n + 6)*x. See Armstrong et al.

Seiichi Manyama, Table of n, a(n) for n = 0..333
D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Row 5 of A306444.

Cf. A000108, A065097 (Cat(n + 1/2)), A265101 (Cat(n + 1/3)), A265102 (Cat(n + 1/4)).

[Binomial(10*n+7, 5*n+1)/(10*n+7): n in [0..15]]; // Vincenzo Librandi, Dec 09 2015

seq(binomial(10*n + 7, 5*n + 1)/(10*n + 7), n = 0..12);

Table[Binomial[10n+7, 5n+1]/(10n+7), {n, 0, 20}] (* Vincenzo Librandi, Dec 09 2015 *)

a(n)=binomial(10*n + 7, 5*n + 1)/(10*n + 7) \\ Anders Hellström, Dec 07 2015

[binomial(10*n+7, 5*n+1)/(10*n+7) for n in (0..20)] # G. C. Greubel, Feb 16 2019

a(n) = binomial(10*n + 7, 5*n + 1)/(10*n + 7).
(n + 1)*(5*n - 2)*(5*n - 3)*(5*n + 4)*(5*n + 6)*a(n) = 32*(2*n + 1)*(10*n + 1)*(10*n - 1)*(10*n + 3)*(10*n - 3)*a(n-1) with a(0) = 1.
From Ilya Gutkovskiy, Feb 28 2017: (Start)
O.g.f.: (5F4(-3/10,-1/10,1/10,3/10,1/2; -3/5,-2/5,4/5,6/5; 1024*x) - 1)/(2*x).
E.g.f.: 5F5(7/10,9/10,11/10,13/10,3/2; 2/5,3/5,9/5,2,11/5; 1024*x).
a(n) ~ 4^(5*n+3)/(5*sqrt(5*Pi)*n^(3/2)). (End)

A306444 A(n,k) = binomial((2k+1)n+2, kn+1)/((2k+1)*n+2), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 7, 1, 1, 14, 66, 30, 1, 1, 42, 715, 1144, 143, 1, 1, 132, 8398, 49742, 22610, 728, 1, 1, 429, 104006, 2340135, 3991995, 482885, 3876, 1, 1, 1430, 1337220, 115997970, 757398510, 347993910, 10855425, 21318, 1

Offset: 0

Seiichi Manyama, Feb 15 2019

			Square array begins:
   1,    1,        1,           1,              1, ...
   1,    2,        5,          14,             42, ...
   1,    7,       66,         715,           8398, ...
   1,   30,     1144,       49742,        2340135, ...
   1,  143,    22610,     3991995,      757398510, ...
   1,  728,   482885,   347993910,   267058714626, ...
   1, 3876, 10855425, 32018897274, 99543581789652, ...

Seiichi Manyama, Antidiagonals n = 0..81, flattened

Columns 0-1 give A000012, A006013.

Rows 0-5 give A000012, A000108(n+1), A065097(n+1), A265101, A265102, A265103.

Flat(List([0..12], n-> List([0..n], k-> Binomial((2*(n-k)+1)*k+2, k*(n-k)+1)/((2*(n-k)+1)*k+2) ))); # G. C. Greubel, Feb 16 2019

[[Binomial((2*(n-k)+1)*k+2, k*(n-k)+1)/((2*(n-k)+1)*k+2): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 16 2019

A[n_, k_]:= Binomial[(2*k+1)*n+2, k*n+1]/((2*k+1)*n+2); Table[A[k, n-k], {n,0,12}, {k,0,n}] (* G. C. Greubel, Feb 16 2019 *)

{A(n,k) = binomial((2*k+1)*n+2, k*n+1)/((2*k+1)*n+2)};
for(n=0,12, for(k=0,n, print1(A(k,n-k), ", "))) \\ G. C. Greubel, Feb 16 2019

[[binomial((2*(n-k)+1)*k+2, k*(n-k)+1)/((2*(n-k)+1)*k+2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Feb 16 2019

A234040 a(n) = binomial(2(n+1),n) gcd(n,2)/(2*(n+1)).

Original entry on oeis.org

1, 1, 5, 7, 42, 66, 429, 715, 4862, 8398, 58786, 104006, 742900, 1337220, 9694845, 17678835, 129644790, 238819350, 1767263190, 3282060210, 24466267020, 45741281820, 343059613650, 644952073662, 4861946401452, 9183676536076, 69533550916004

Offset: 0

Wolfdieter Lang, Feb 23 2014

This gives the next-to-central entries of the even-indexed rows of the triangle A107711.
For the central entries (of the even-numbered rows) see A001700.
This sequence is composed of the bisection sequences A024492 (even part) and A065097 (odd part).

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

Cf. A000108, A024492, A065097, A107711.

[Binomial(2*(n+1),n)*Gcd(n,2)/(2*(n+1)): n in [0..30]]; // Vincenzo Librandi, Feb 25 2014

Table[Binomial[2 (n + 1), n] GCD[n, 2]/(2 (n + 1)), {n, 0, 40}] (* Vincenzo Librandi, Feb 25 2014 *)

a(n) = binomial(2*(n+1),n)*gcd(n,2)/(2*(n+1)) for n >= 0.
a(n) = A107711(2*(n+1), n) for n >= 0.
G.f.: (3*c(x)- c(-x)-2)/(4*x) =(4*(1-x) - 3*sqrt(1-4*x) - sqrt(1+4*x))/(8*x^2), with c(x) the o.g.f. of the Catalan numbers A000108. See the bisection comment above.

a(26) from Vincenzo Librandi, Feb 25 2014

A349740 Number of partitions of set [n] in a set of <= k noncrossing subsets. Number of Dyck n-paths with at most k peaks. Both with 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 7, 13, 14, 0, 1, 11, 31, 41, 42, 0, 1, 16, 66, 116, 131, 132, 0, 1, 22, 127, 302, 407, 428, 429, 0, 1, 29, 225, 715, 1205, 1401, 1429, 1430, 0, 1, 37, 373, 1549, 3313, 4489, 4825, 4861, 4862, 0, 1, 46, 586, 3106, 8398, 13690, 16210, 16750, 16795, 16796

Offset: 0

Ron L.J. van den Burg, Nov 28 2021

Given a partition P of the set {1,2,...,n}, a crossing in P are four integers [a, b, c, d] with 1 <= a < b < c < d <= n for which a, c are together in a block, and b, d are together in a different block. A noncrossing partition is a partition with no crossings.

			For n=4 the T(4,3)=13 partitions are {{1,2,3,4}}, {{1,2,3},{4}}, {{1,2,4},{3}}, {{1,3,4},{2}}, {{2,3,4},{1}}, {{1,2},{3,4}}, {{1,4},{2,3}}, {{1,2},{3},{4}}, {{1,3},{2},{4}}, {{1,4},{2},{3}}, {{1},{2,3},{4}}, {{1},{2,4},{3}}, {{1},{2},{3,4}}.
The set of sets {{1,3},{2,4}} is missing because it is crossing. If you add the set of 4 sets, {{1},{2},{3},{4}}, you get T(4, 4) = 14 = A000108(4), the 4th Catalan number.
Triangle begins:
  1;
  0, 1;
  0, 1,  2;
  0, 1,  4,   5;
  0, 1,  7,  13,   14;
  0, 1, 11,  31,   41,   42;
  0, 1, 16,  66,  116,  131,  132;
  0, 1, 22, 127,  302,  407,  428,  429;
  0, 1, 29, 225,  715, 1205, 1401, 1429, 1430;
  0, 1, 37, 373, 1549, 3313, 4489, 4825, 4861, 4862;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
David Callan, Sets, Lists and Noncrossing Partitions, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.3; also on arXiv, arXiv:0711.4841 [math.CO], 2007-2008.

Columns k=0-4 give (for n>=k): A000007, A000012, A000124(n-1), A116701, A116844.
Partial sums of A090181 per row.
Main diagonal is A000108.
Row sums give A088218.
T(2*n,n) gives A065097.
T(n,n-1) gives A001453 for n >= 2.
Cf. A106396, A137940.

b:= proc(x, y, t) option remember; expand(`if`(y<0
      or y>x, 0, `if`(x=0, 1, add(b(x-1, y+j, j)*
     `if`(t=1 and j<1, z, 1), j=[-1, 1]))))
    end:
T:= proc(n, k) option remember; `if`(k<0, 0,
      T(n, k-1)+coeff(b(2*n, 0$2), z, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Nov 28 2021

T[n_, k_] := If[n == 0, 1, Sum[j Binomial[n, j]^2 / (n - j + 1), {j, 0, k}] / n];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Peter Luschny, Nov 29 2021 *)

T(n,k) = Sum_{j=0..k} A090181(n,j), the partial sum of the Narayana numbers.
T(n,n) = A000108(n), the n-th Catalan number.
G.f.: (1 + x - x*y - sqrt((1-x*(1+y))^2 - 4*y*x^2))/(2*x*(1-y)).
T(n,k) = (1/n)*Sum_{j=0..k} j*binomial(n,j)^2 / (n-j+1) for n >= 1. - Peter Luschny, Nov 29 2021

A065424 Catalan-like formula: a(n) = binomial(6n, 3n+1)/(9*n+6).

Original entry on oeis.org

1, 33, 1326, 59432, 2851425, 143291610, 7446255180, 396893583792, 21579377870484, 1192183281903845, 66734212415276406, 3776778437640143208, 215744630060724034270, 12423227699242323077940, 720356761939547257421400, 42024927437494196952957408, 2464931252806478840545733484

Offset: 1

Len Smiley, Nov 16 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A000108, A000245, A065097.

[Binomial(6*n,3*n+1)/(9*n+6): n in [1..20]]; // Vincenzo Librandi, Nov 17 2011

a[n_] := Binomial[6*n, 3*n+1]/(9*n+6); Array[a, 20] (* Amiram Eldar, Sep 04 2025 *)

G.f.: A*sqrt((A+1)*(1+9*A)) where A=x*(1+9*A)^3*(1+A). - Mark van Hoeij, Nov 16 2011
-(n-1)*(3*n+2)*(3*n+1)*a(n) + 8*(6*n-5)*(6*n-1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 31 2015
a(n) = (1/9)*A000245(3*n) = n*A000108(3*n)/(3*n + 2) for n >= 1. - Peter Bala, Mar 08 2023
a(n) ~ 2^(6*n) / (9 * sqrt(3*Pi) * n^(3/2)). - Amiram Eldar, Sep 04 2025

A358723 Number of n-node rooted trees of edge-height equal to their number of leaves.

Original entry on oeis.org

0, 1, 0, 2, 1, 6, 7, 26, 43, 135, 276, 755, 1769, 4648, 11406, 29762, 75284, 195566, 503165, 1310705, 3402317, 8892807, 23231037, 60906456, 159786040, 420144405, 1105673058, 2914252306, 7688019511, 20304253421, 53667498236, 141976081288, 375858854594, 995728192169

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Edge-height (A109082) is the number of edges in the longest path from root to leaf.

			The a(1) = 0 through a(7) = 7 trees:
  .  (o)  .  ((oo))  ((o)(o))  (((ooo)))  (((o))(oo))
             (o(o))            ((o(oo)))  (((o)(oo)))
                               ((oo(o)))  ((o)((oo)))
                               (o((oo)))  ((o)(o(o)))
                               (o(o(o)))  ((o(o)(o)))
                               (oo((o)))  (o((o)(o)))
                                          (o(o)((o)))

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

For internals instead of leaves: A011782, ranked by A209638.
For internals instead of edge-height: A185650 aerated, ranked by A358578.
For node-height: A358589 (square trees), ranked by A358577, ordered A358590.
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 internals, ordered A090181.
Cf. A065097, A109082, A109129, A342507, A358552, A358587, A358591, A358728.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{-2}]==Depth[#]-2&]],{n,1,10}]

\\ Needs R(n,f) defined in A358589.
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h-1,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023

cp's OEIS Frontend

A265102 a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6).

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A265103 a(n) = binomial(10*n + 7, 5*n + 1)/(10*n + 7).

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A306444 A(n,k) = binomial((2*k+1)*n+2, k*n+1)/((2*k+1)*n+2), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

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A234040 a(n) = binomial(2*(n+1),n) * gcd(n,2)/(2*(n+1)).

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A349740 Number of partitions of set [n] in a set of <= k noncrossing subsets. Number of Dyck n-paths with at most k peaks. Both with 0 <= k <= n, read by rows.

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Maple

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A065424 Catalan-like formula: a(n) = binomial(6*n, 3*n+1)/(9*n+6).

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Magma

Mathematica

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A358723 Number of n-node rooted trees of edge-height equal to their number of leaves.

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A265102 a(n) = binomial(8n + 6, 4n + 1)/(8*n + 6).

A265103 a(n) = binomial(10n + 7, 5n + 1)/(10*n + 7).

A306444 A(n,k) = binomial((2k+1)n+2, kn+1)/((2k+1)*n+2), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

A234040 a(n) = binomial(2(n+1),n) gcd(n,2)/(2*(n+1)).

A065424 Catalan-like formula: a(n) = binomial(6n, 3n+1)/(9*n+6).