A065097 -id:A065097 - cp's OEIS frontend

A000891 a(n) = (2n)!(2n+1)! / (n! (n+1)!)^2.

1, 3, 20, 175, 1764, 19404, 226512, 2760615, 34763300, 449141836, 5924217936, 79483257308, 1081724803600, 14901311070000, 207426250094400, 2913690606794775, 41255439318353700, 588272005095043500

Offset: 0

N. J. A. Sloane

nonn

Number of parallelogram polyominoes having n+1 columns and n+1 rows. - Emeric Deutsch, May 21 2003
Number of tilings of an  hexagon.
a(n) is the number of non-crossing partitions of [2n+1] into n+1 blocks. For example, a[1] counts 13-2, 1-23, 12-3. - David Callan, Jul 25 2005
The number of returning walks of length 2n on the upper half of a square lattice, since a(n) = Sum_{k=0..2n} binomial(2n,k)*A126120(k)*A126869(n-k). - Andrew V. Sutherland, Mar 24 2008
For sequences counting walks in the upper half-plane starting from the origin and finishing at the lattice points (0,m) see A145600 (m = 1), A145601 (m = 2), A145602 (m = 3) and A145603 (m = 4). - Peter Bala, Oct 14 2008
The number of proper mergings of two n-chains. - Henri Mühle, Aug 17 2012
a(n) is number of pairs of non-intersecting lattice paths from (0,0) to (n+1,n+1) using (1,0) and (0,1) as steps. Here, non-intersecting means two paths do not share a vertex except the origin and the destination. For example, a(1) = 3 because we have three such pairs from (0,0) to (2,2): {NNEE,EENN}, {NNEE,ENEN}, {NENE,EENN}. - Ran Pan, Oct 01 2015
Also the number of ordered rooted trees with 2(n+1) nodes and n+1 leaves, i.e., half of the nodes are leaves. These trees are ranked by A358579. The unordered version is A185650. - Gus Wiseman, Nov 27 2022
The number of secondary GL(2) invariants constructed from n+1 two component vectors. This number was evaluated by using the Molien-Weyl formula to compute the Hilbert series of the ring of invariants. - Jaco van Zyl, Jun 30 2025

			G.f. = 1 + 3*x + 20*x^2 + 175*x^3 + 1764*x^4 + 19404*x^5 + ...
From _Gus Wiseman_, Nov 27 2022: (Start)
The a(2) = 20 ordered rooted trees with 6 nodes and 3 leaves:
  (((o)oo))  (((o)o)o)  (((o))oo)
  (((oo)o))  (((oo))o)  ((o)(o)o)
  (((ooo)))  ((o)(oo))  ((o)o(o))
  ((o(o)o))  ((o(o))o)  (o((o))o)
  ((o(oo)))  ((oo)(o))  (o(o)(o))
  ((oo(o)))  (o((o)o))  (oo((o)))
             (o((oo)))
             (o(o(o)))
(End)

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 94.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
Elena Barcucci, Andrea Frosini and Simone Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011), Article 11.4.5.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, arXiv:0804.2930 [math.CO], 2008.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, Discrete Math., 309 (2009), 2834-2838.
Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.
Ivan Marin and Emmanuel Wagner, A cubic defining algebra for the Links-Gould polynomial. arXiv preprint arXiv:1203.5981 [math.GT], 2012. - From _N. J. A. Sloane_, Sep 21 2012
Henri Mühle, Counting Proper Mergings of Chains and Antichains, arXiv:1206.3922 [math.CO], 2012.
Guoce Xin, Determinant formulas relating to tableaux of bounded height, Adv. Appl. Math. 45 (2010) 197-211.

Cf. A000356, A010370, A038535.
Cf. A145600, A145601, A145602, A145603. - Peter Bala, Oct 14 2008
Cf. A000142, A000894, A248045.
Equals half of A267981.
Counts the trees ranked by A358579.
A001263 counts ordered rooted trees by nodes and leaves.
A090181 counts ordered rooted trees by nodes and internals.
Cf. A000108, A065097, A080936, A185650, A358371, A358586, A358590.

a000891 n = a001263 (2 * n - 1) n  -- Reinhard Zumkeller, Oct 10 2013

[Factorial(2*n)*Factorial(2*n+1) / (Factorial(n) * Factorial(n+1))^2: n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

with(combstruct): bin := {B=Union(Z,Prod(B,B))} :seq(1/2*binomial(2*i,i)*(count([B,bin,unlabeled],size=i)), i=1..18) ; # Zerinvary Lajos, Jun 06 2007

a[ n_] := If[ n == -1, 0, Binomial[2 n + 1, n]^2 / (2 n + 1)]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (1 - Hypergeometric2F1[ -1/2, 1/2, 1, 16 x]) / (4 x), {x, 0, n}]; (* Michael Somos, May 28 2014 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ BesselI[0, 2 x] BesselI[1, 2 x] / x, {x, 0, 2 n}]]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticE[ 16 x] / (Pi/2)) / (4 x), {x, 0, n}]; (* Michael Somos, Sep 18 2016 *)
a[n_] := (2 n + 1) CatalanNumber[n]^2;
Array[a, 20, 0] (* Peter Luschny, Mar 03 2020 *)

{a(n) = binomial(2*n+1, n)^2 / (2*n + 1)}; /* Michael Somos, Jun 22 2005 */

a(n) = matdet(matrix(n, n, i, j, binomial(n+j+1,i+1))) \\ Hugo Pfoertner, Oct 22 2022

-4*a(n) = A010370(n+1).
G.f.: (1 - E(16*x)/(Pi/2))/(4*x) where E() is the elliptic integral of the second kind. [edited by Olivier Gérard, Feb 16 2011]
G.f.: 3F2(1, 1/2, 3/2; 2,2; 16*x)= (1 - 2F1(-1/2, 1/2; 1; 16*x)) / (4*x). - Olivier Gérard, Feb 16 2011
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2*x) * BesselI(1, 2*x) / x. - Michael Somos, Jun 22 2005
a(n) = A001700(n)*A000108(n) = (1/2)*A000984(n+1)*A000108(n). - Zerinvary Lajos, Jun 06 2007
For n > 0, a(n) = (n+2)*A000356(n) starting (1, 5, 35, 294, ...). - Gary W. Adamson, Apr 08 2011
a(n) = A001263(2*n+1,n+1) = binomial(2*n+1,n+1)*binomial(2*n+1,n)/(2*n+1) (central members of odd numbered rows of Narayana triangle).
G.f.: If G_N(x) = 1 + Sum_{k=1..N} ((2*k)!*(2*k+1)!*x^k)/(k!*(k+1)!)^2, G_N(x) = 1 + 12*x/(G(0) - 12*x); G(k) = 16*x*k^2 + 32*x*k + k^2 + 4*k + 12*x + 4 - 4*x*(2*k+3)*(2*k+5)*(k+2)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
D-finite with recurrence (n+1)^2*a(n) - 4*(2*n-1)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
a(n) = A005558(2n). - Mark van Hoeij, Aug 20 2014
a(n) = A000894(n) / (n+1) = A248045(n+1) / A000142(n+1). - Reinhard Zumkeller, Sep 30 2014
From Ilya Gutkovskiy, Feb 01 2017: (Start)
E.g.f.: 2F2(1/2,3/2; 2,2; 16*x).
a(n) ~ 2^(4*n+1)/(Pi*n^2). (End)
a(n) = A005408(n)*(A000108(n))^2. - Ivan N. Ianakiev, Nov 13 2019
a(n) = det(M(n)) where M(n) is the n X n matrix with m(i,j) = binomial(n+j+1,i+1). - Benoit Cloitre, Oct 22 2022
a(n) = Integral_{x=0..16} x^n*W(x) dx, where W(x) = (16*EllipticE(1 - x/16) - x*EllipticK(1 - x/16))/(8*Pi^2*sqrt(x)), n=>0. W(x) diverges at x=0, monotonically decreases for x>0, and vanishes at x=16. EllipticE and EllipticK are elliptic functions. This integral representation as n-th moment of a positive function W(x) on the interval [0, 16] is unique. - Karol A. Penson, Dec 20 2023

More terms from Andrew V. Sutherland, Mar 24 2008

A028364 Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 5, 7, 9, 14, 14, 19, 23, 28, 42, 42, 56, 66, 76, 90, 132, 132, 174, 202, 227, 255, 297, 429, 429, 561, 645, 715, 785, 869, 1001, 1430, 1430, 1859, 2123, 2333, 2529, 2739, 3003, 3432, 4862, 4862, 6292, 7150, 7810, 8398, 8986, 9646, 10504, 11934, 16796

Offset: 0

Wouter Meeussen

There are several versions of a Catalan triangle: see A009766, A008315, A028364.
The subtriangle [1], [2, 3], [5, 7, 9], ..., namely T(N,M-1), for N >= 1, M=1..N, appears as one-point function in the totally asymmetric exclusion process for the parameters alpha=1=beta. See the Derrida et al. and Liggett references given under A067323, where these triangle entries are called T_{N,N+M-1} for the given alpha and beta values. See the row reversed triangle A067323.
Consider a Dyck path as a path with steps N=(0,1) and E=(1,0) from (0,0) to (n,n) that stays weakly above y=x. T(n,m) is the number of Dyck paths of semilength n+1 where the (m+1)st north step is followed by an east step. - Lara Pudwell, Apr 12 2023

			Triangle begins
   1;
   1,  2;
   2,  3,  5;
   5,  7,  9, 14;
  14, 19, 23, 28, 42;

Alois P. Heinz, Rows n = 0..140, flattened
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
G. Chatel and V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.

Cf. A000108 (column 0 and main diagonal), A001700 (row sums), A065097 (T(2*n-1, n-1)), A201205 (central terms).

Cf. A067323, A009766, A039598, A039599, A028377, A028378, A028376.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b((n+1)$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 28 2015

t[n_, k_] = Sum[CatalanNumber[n-j]*CatalanNumber[j], {j, 0, k}]; Flatten[Table[t[n, k], {n, 0, 8}, {k, 0, n}]] (* Jean-François Alcover, Jul 22 2011 *)

T(n,k) = Sum_{j>=0} A039598(k,j)*A039599(n-k,j). - Philippe Deléham, Feb 18 2004
Sum_{k>=0} T(n,k) = A001700(n). T(n,k) = A067323(n,n-k), n >= k >= 0, otherwise 0. - Philippe Deléham, May 26 2005
G.f. for column sequences m >= 0: (-(c(m,x)-1)/x+c(m,x)*c(x))/x^m with the g.f. c(x) of A000108 (Catalan) and c(m,x):=sum(C(k)*x^k,k=0..m) with C(n):=A000108(n). - Wolfdieter Lang, Mar 24 2006
G.f. for column sequences m >= 0 (without leading zeros): c(x)*Sum_{k=0..m} C(m,k)*c(x)^k with the g.f. c(x) of A000108 (Catalan) and C(n,m) is the Catalan triangle A033184(n,m). - Wolfdieter Lang, Mar 24 2006
T(n,n) = T(n,k) + T(n,n-1-k) = A000108(n+1), n > 0, k = 0..floor((n+1)/2). - Yuchun Ji, Jan 09 2019
G.f. for triangle: Sum_{n>=0, m>=0} T(n, m)*x^n*y^m = (c(x)-c(xy))/(x(1-y)c(x)) with the g.f. c(x) of A000108 (Catalan). - Lara Pudwell, Apr 12 2023

A048990 Catalan numbers with even index (A000108(2n), n >= 0): a(n) = binomial(4n, 2n)/(2n+1).

Original entry on oeis.org

1, 2, 14, 132, 1430, 16796, 208012, 2674440, 35357670, 477638700, 6564120420, 91482563640, 1289904147324, 18367353072152, 263747951750360, 3814986502092304, 55534064877048198, 812944042149730764, 11959798385860453492, 176733862787006701400

Offset: 0

Wolfdieter Lang

With interpolated zeros, this is C(n)*(1+(-1)^n)/2 with g.f. given by 2/(sqrt(1+4x) + sqrt(1-4x)). - Paul Barry, Sep 09 2004
Self-convolution of a(n)/4^n gives Catalan numbers (A000108). - Vladimir Reshetnikov, Oct 10 2016
a(n) is the number of grand Dyck paths from (0,0) to (4n,0) that avoid vertices (2k,0) for all odd k > 0. - Alexander Burstein, May 11 2021
a(n) is the number of lattice paths from (0,0) to (2n,2n) with steps (1,0) and (0,1) that avoid the points (1,1), (3,3), (5,5), ..., (2n-1,2n-1).  This is Example 2.5 of the Shapiro reference. - Lucas A. Brown, Jul 24 2025

			sqrt(2*x^-1*(1-sqrt(1-x))) = 1 + (1/8)*x + (7/128)*x^2 + (33/1024)*x^3 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Vol. 491 (2016), pp. 123-137.
Greg Markowsky, A method for deriving hypergeometric and related identities from the H 2 Hardy norm of conformal maps, Integral Transforms and Special Functions, Vol. 24, No. 4 (2013), pp. 302-313; arXiv preprint, arXiv:1205.2458 [math.CV], 2012.
Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood, Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients, J. Int. Seq., Vol. 18 (2015), Article 15.11.7.
Louis Shapiro, Random walks with absorbing points, Discrete Applied Mathematics, Vol. 39 (1992), pp. 57-67.

Cf. A000007, A000108, A024492, A065097, A099250, A187357, A187358, A187359, A228411, A276483.

Cf. A000984, A078531, A182122, A245112, A245113.

a[n_] := CatalanNumber[2n]; Array[a, 18, 0] (* Or *)
CoefficientList[ Series[ Sqrt[2]/Sqrt[1 + Sqrt[1 - 16 x]], {x, 0, 17}], x] (* Robert G. Wilson v *)
CatalanNumber[Range[0,40,2]] (* Harvey P. Dale, Mar 19 2015 *)

combinat::dyckWords::count(2*n) $ n = 0..28 // Zerinvary Lajos, Apr 14 2007

/* G.f.: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ): */
{a(n)=local(A=1+x); for(i=1, n, A=exp(x*A^4 + intformal(A^4 +x*O(x^n)))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 09 2013
for(n=0, 30, print1(a(n), ", "))

A048990 = lambda n: hypergeometric([1-2*n,-2*n],[2],1)
[Integer(A048990(n).n()) for n in range(20)] # Peter Luschny, Sep 22 2014

a(n) = 2 * A065097(n) - A000007(n).
G.f.: A(x) = sqrt((1/8)*x^(-1)*(1-sqrt(1-16*x))).
G.f.: 2F1( (1/4, 3/4); (3/2))(16*x). - Olivier Gérard Feb 17 2011
D-finite with recurrence n*(2*n+1)*a(n) - 2*(4*n-1)*(4*n-3)*a(n-1) = 0. - R. J. Mathar, Nov 30 2012
E.g.f: 2F2(1/4, 3/4; 1, 3/2; 16*x). - Vladimir Reshetnikov, Apr 24 2013
G.f. A(x) satisfies: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ). - Paul D. Hanna, Nov 09 2013
G.f. A(x) satisfies: A(x) = sqrt(1 + 4*x*A(x)^4). - Paul D. Hanna, Nov 09 2013
a(n) = hypergeom([1-2*n,-2*n],[2],1). - Peter Luschny, Sep 22 2014
a(n) ~ 2^(4*n-3/2)/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Oct 10 2016
From Peter Bala, Feb 27 2020: (Start)
a(n) = (4^n)*binomial(2*n + 1/2, n)/(4*n + 1).
O.g.f.: A(x) = sqrt(c(4*x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers. Cf. A228411. (End)
Sum_{n>=0} 1/a(n) = A276483. - Amiram Eldar, Nov 18 2020
Sum_{n>=0} a(n)/4^n = sqrt(2). - Amiram Eldar, Mar 16 2022
From Peter Bala, Feb 22 2023: (Start)
a(n) = (1/2^(2*n-1)) * Product_{1 <= i <= j <= 2*n-1} (i + j + 2)/(i + j - 1) for n >= 1.
a(n) = Product_{1 <= i <= j <= 2*n-1} (3*i + j + 2)/(3*i + j - 1). Cf. A024492. (End)
a(n) = Sum_{k = 0..2*n-1} (-1)^k * 4^(2*n-k-1)*binomial(2*n-1, k)*Catalan(k+1). - Peter Bala, Apr 29 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^6). - Seiichi Manyama, Jun 20 2025

A358589 Number of square rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 11, 17, 55, 107, 317, 720, 1938, 4803, 12707, 32311, 85168, 220879, 581112, 1522095, 4014186, 10568936, 27934075, 73826753, 195497427, 517927859, 1373858931, 3646158317, 9684878325, 25737819213, 68439951884, 182070121870, 484583900955, 1290213371950

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

We say that a tree is square if it has the same height as number of leaves.

			The a(1) = 1 through a(7) = 11 trees:
  o  .  (oo)  .  ((ooo))  ((o)(oo))  (((oooo)))
                 (o(oo))  (o(o)(o))  ((o(ooo)))
                 (oo(o))             ((oo(oo)))
                                     ((ooo(o)))
                                     (o((ooo)))
                                     (o(o(oo)))
                                     (o(oo(o)))
                                     (oo((oo)))
                                     (oo(o(o)))
                                     (ooo((o)))
                                     ((o)(o)(o))

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

For internals instead of height we have A185650 aerated, ranked by A358578.
These trees are ranked by A358577.
For internals instead of leaves we have A358587, ranked by A358576.
The ordered version is 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 internal nodes, ordered A090181.
Cf. A000891, A065097, A109129, A358552, A358588, 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}]==Depth[#]-1&]],{n,1,10}]

\\ R(n,f) enumerates trees by height(h), nodes(x) and leaves(y).
R(n,f) = {my(A=O(x*x^n), Z=0); for(h=1, n, my(p = A); A = x*(y - 1  + exp( sum(i=1, n-1, 1/i * subst( subst( A + O(x*x^((n-1)\i)), x, x^i), y, y^i) ) )); Z += f(h, A-p)); Z}
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

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

A358590 Number of square ordered rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 6, 5, 36, 84, 309, 890, 3163, 9835, 32979, 108252, 360696, 1192410, 3984552, 13276769, 44371368, 148402665, 497072593, 1665557619, 5586863093, 18750662066, 62968243731, 211565969511, 711187790166, 2391640404772, 8045964959333, 27077856222546

Offset: 1

Gus Wiseman, Nov 25 2022

nonn

We say that a tree is square if it has the same height as number of leaves.

			The a(1) = 1 through a(6) = 5 ordered trees:
  o  .  (oo)  .  ((o)oo)  ((o)(o)o)
                 ((oo)o)  ((o)(oo))
                 ((ooo))  ((o)o(o))
                 (o(o)o)  ((oo)(o))
                 (o(oo))  (o(o)(o))
                 (oo(o))

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

For internals instead of height we have A000891, unordered A185650 aerated.
For internals instead of leaves we have A358588, unordered A358587.
The unordered version is A358589, ranked by A358577.
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, A109129, A358371, A358552, A358579, A358586.

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}]

\\ R(n,f) enumerates trees by height(h), nodes(x) and leaves(y).
R(n,f) = {my(A=O(x*x^n), Z=0); for(h=1, n, my(p = A); A = x*(y - 1  + 1/(1 - A + O(x^n))); Z += f(h, A-p)); Z}
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

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

A001794 Negated coefficients of Chebyshev T polynomials: [x^n](-T(n+6, x)), n >= 0.

Original entry on oeis.org

1, 7, 32, 120, 400, 1232, 3584, 9984, 26880, 70400, 180224, 452608, 1118208, 2723840, 6553600, 15597568, 36765696, 85917696, 199229440, 458752000, 1049624576, 2387607552, 5402263552, 12163481600, 27262976000, 60850962432

Offset: 0

N. J. A. Sloane

A negated subdiagonal of A053120.
If X_1,X_2,...,X_n are 2-blocks of a (2n+1)-set X then a(n-2) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
The third corrector line for transforming 2^n offset 0 with a leading 1 into the Fibonacci sequence. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
a(n-2) is the number of strings of length n defined on {0, 1, 2, 3} that have exactly two 2's and no 3's or exactly three 3's and no 2's. For example, for n=4, a(2)=32 since the strings are the 6 permutations of 2200, the 12 permutations of 2201, the 6 permutations of 2211, the 4 permutations of 3330, and the 4 permutations of 3331. - Enrique Navarrete, Jun 03 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
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).

T. D. Noe, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Michael Albert, Mike Atkinson and Robert Brignall, The enumeration of three pattern classes using monotone grid classes, E. J. Combinat., Vol. 19, No. 3 (2012), Article P20, chapter 5.4.1.
Takayuki Hibi, Nan Li and Hidefumi Ohsugi, The face vector of a half-open hypersimplex, arXiv preprint arXiv:1309.5155 [math.CO], 2013-2014.
Milan Janjic, Two Enumerative Functions.
C. W. Jones, J. C. P. Miller, J. F. C. Conn, and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
John Machacek, A finite totally nonnegative Grassmannian, arXiv:2410.06177 [math.CO], 2024. See p. 11.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A039991 (negative of column 6), A028297, A008310, A053120.
With alternating signs, the o.g.f. (with offset 1) is the inverse of the o.g.f. of A065097.
Cf. A001789 (partial sums), A081279 (binomial transform), A005900 (0 followed by inverse binomial transform).

List([0..25],n->2^(n-2)*(n+1)*(n+2)*(n+6)/3); # Muniru A Asiru, Mar 20 2018

[2^(n-1)/3*Binomial(n+2,2)*(n+6) : n in [0..25]]; // Brad Clardy, Mar 08 2012

[seq(coeftayl((1-x)/(1-2*x)^4, x = 0, k), k=0..25)]; # Muniru A Asiru, Mar 20 2018

a[n_] := 2^(n-2)*(n+1)*(n+2)*(n+6)/3; a /@ Range[0, 20] (* Giovanni Resta, Mar 25 2017 *)
LinearRecurrence[{8,-24,32,-16},{1,7,32,120},30] (* Harvey P. Dale, Oct 08 2024 *)

a(n) = sum(i=0, n+1, sum(k=0, i, k^2*binomial(n+1, i))); \\ Michel Marcus, Mar 25 2017

a(n) = - polcoeff(polchebyshev(n+6), n); \\ Michel Marcus, Mar 20 2018

a(n) = 2^(n-2)*(n+1)*(n+2)*(n+6)/3. [See a comment in A053120 on subdiagonals. - Wolfdieter Lang, Jan 03 2020]
G.f.: (1-x)/(1-2*x)^4. - Simon Plouffe in his 1992 dissertation
a(n) = Sum_{k=0..floor((n+6)/2)} C(n+6, 2*k)*C(k, 3). - Paul Barry, May 15 2003
With a leading zero, the binomial transform of A000330. - Paul Barry, Jul 19 2003
a(n) = Sum_{i=0..n+1} (Sum{k=0..i} (k^2*binomial(n+1, i))). - Jon Perry, Feb 26 2004 [corrected by Michel Marcus, Mar 25 2017]
Binomial transform of a(n) = (2*n^3 + 6*n^2 + 7*n + 3)/3 offset 0. a(3)=120. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
a(n) = (2^(n-1)/3)*binomial(n+2,2)*(n+6). - Brad Clardy, Mar 08 2012
E.g.f.: (1/3)*exp(2*x)*(3 + 15*x + 12*x^2 + 2*x^3). - Stefano Spezia, Jan 03 2020
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 156*log(2)/5 - 511/25.
Sum_{n>=0} (-1)^n/a(n) = 241/25 - 108*log(3/2)/5. (End)
E.g.f.: exp(2*x)*(x^2/2 + x^3/6) (with two leading zeros). - Enrique Navarrete, Jun 03 2025

Name clarified by Wolfdieter Lang, Nov 26 2019

A358586 Number of ordered rooted trees with n nodes, at least half of which are leaves.

Original entry on oeis.org

1, 1, 1, 4, 7, 31, 66, 302, 715, 3313, 8398, 39095, 104006, 484706, 1337220, 6227730, 17678835, 82204045, 238819350, 1108202513, 3282060210, 15195242478, 45741281820, 211271435479, 644952073662, 2971835602526, 9183676536076, 42217430993002, 131873975875180, 604834233372884

Offset: 1

Gus Wiseman, Nov 24 2022

nonn

			The a(1) = 1 through a(5) = 7 ordered trees:
  o  (o)  (oo)  (ooo)   (oooo)
                ((o)o)  ((o)oo)
                ((oo))  ((oo)o)
                (o(o))  ((ooo))
                        (o(o)o)
                        (o(oo))
                        (oo(o))

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

For equality we have A000891, unordered A185650.
Odd-indexed terms appear to be A065097.
The unordered version is A358583.
The opposite is the same, unordered A358584.
The strict case is A358585, unordered A358581.
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, A358582, A358584, A358588.

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,1,10}]

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/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

A358587 Number of n-node rooted trees of height equal to the number of internal (non-leaf) nodes.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 14, 41, 111, 282, 688, 1627, 3761, 8540, 19122, 42333, 92851, 202078, 436916, 939359, 2009781, 4281696, 9087670, 19223905, 40544951, 85284194, 178956984, 374691171, 782936761, 1632982372, 3400182458, 7068800357, 14674471611, 30422685030

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

			The a(5) = 1 through a(7) = 14 trees:
  ((o)(o))  ((o)(oo))   ((o)(ooo))
            (o(o)(o))   ((oo)(oo))
            (((o)(o)))  (o(o)(oo))
            ((o)((o)))  (oo(o)(o))
                        (((o))(oo))
                        (((o)(oo)))
                        ((o)((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 leaves instead of height we have A185650 aerated, ranked by A358578.
These trees are ranked by A358576.
The ordered version is A358588.
Square trees are counted by A358589, 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 internal nodes, ordered A090181.
Cf. A000891, A065097, A342507, A358552, A358581-A358584, 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}]==Depth[#]-1&]],{n,1,10}]

\\ Needs R(n,f) defined in A358589.
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 15 2024: (Start)
a(n) = 5*a(n-1) - 7*a(n-2) - a(n-3) + 8*a(n-4) - 4*a(n-5) for n > 7.
G.f.: x^5*(x^2 - x + 1)/((x - 1)^2*(x + 1)*(2*x - 1)^2). (End)

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

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

Original entry on oeis.org

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

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}]

cp's OEIS Frontend

A000891 a(n) = (2*n)!*(2*n+1)! / (n! * (n+1)!)^2.

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A028364 Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).

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A048990 Catalan numbers with even index (A000108(2*n), n >= 0): a(n) = binomial(4*n, 2*n)/(2*n+1).

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A358589 Number of square rooted trees with n nodes.

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A358590 Number of square ordered rooted trees with n nodes.

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A001794 Negated coefficients of Chebyshev T polynomials: [x^n](-T(n+6, x)), n >= 0.

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A000891 a(n) = (2n)!(2n+1)! / (n! (n+1)!)^2.

A048990 Catalan numbers with even index (A000108(2n), n >= 0): a(n) = binomial(4n, 2n)/(2n+1).