A003645 -id:A003645 - cp's OEIS frontend

A201639 Triangle read by rows, T(n,k) for 0<=k<=n, generalizes the Motzkin lattice paths with weights of A003645.

1, 4, 1, 20, 8, 1, 112, 56, 12, 1, 672, 384, 108, 16, 1, 4224, 2640, 880, 176, 20, 1, 27456, 18304, 6864, 1664, 260, 24, 1, 183040, 128128, 52416, 14560, 2800, 360, 28, 1, 1244672, 905216, 396032, 121856, 27200, 4352, 476, 32, 1, 8599552, 6449664, 2976768

Offset: 0

Peter Luschny, Sep 20 2012

			[0] [1]
[1] [4, 1]
[2] [20, 8, 1]
[3] [112, 56, 12, 1]
[4] [672, 384, 108, 16, 1]
[5] [4224, 2640, 880, 176, 20, 1]
[6] [27456, 18304, 6864, 1664, 260, 24, 1]
[7] [183040, 128128, 52416, 14560, 2800, 360, 28, 1]

Muniru A Asiru, Table of n, a(n) for n = 0..9044

Sum of row n is A194723(n+1).

Cf. A003645.

Flat(List([0..10], n->List([0..n],k->(k+1)*2^(n-k)*Binomial(2*(n+1),n-k)/(n+1)))); # Muniru A Asiru, Apr 07 2018

/* As triangle */ [[(k+1)*2^(n-k)*Binomial(2*(n+1),n-k)/(n+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Apr 07 2018

Flatten[Table[(k + 1) 2^(n - k) Binomial[2 (n + 1), n - k] / (n + 1), {n, 0, 11}, {k, 0, n}]] (* Vincenzo Librandi, Apr 07 2018 *)

T(n,k) = (k+1)*2^(n-k)*binomial(2*(n+1),n-k)/(n+1);
tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Apr 07 2018

def A201639_triangle(dim):
    T = matrix(ZZ,dim,dim)
    for n in range(dim): T[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            T[n,k] = T[n-1,k-1]+4*T[n-1,k]+4*T[n-1,k+1]
    return T
A201639_triangle(9)

Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1)+4*T(n-1,k)+4*T(n-1,k+1).
G.f.: -(4*x+sqrt(1-8*x)-1)/((4*x^2-x)*y+sqrt(1-8*x)*x*y+8*x^2). - Vladimir Kruchinin, Apr 06 2018
T(n,k) = (k+1)*2^(n-k)*C(2*(n+1),n-k)/(n+1). - Vladimir Kruchinin, Apr 06 2018

A052701 a(0) = 0; for n >= 1, a(n) = 2^(n-1)*C(n-1), where C(n) = A000108(n) Catalan numbers, n>0.

Original entry on oeis.org

0, 1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, 17199104, 120393728, 852017152, 6085836800, 43818024960, 317680680960, 2317200261120, 16992801914880, 125210119372800, 926554883358720, 6882979133521920

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A151374 shifted one place right. - Joerg Arndt, Mar 17 2011
The number of rooted Eulerian n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
This is also the number of strings of length 2n-2 of two different types of balanced parentheses. For example, a(2) = 2, since the two possible strings of length 2 are [] and (), a(3) = 8, since the 8 possible strings of length 4 are (()), [()], ([]), [[]], ()(), [](), ()[], and [][]. - Jeffrey Shallit, Jun 03 2006
Row sums of number triangle A110506. - Paul Barry, Jul 24 2005
Also row sums of triangle in A085880. - Philippe Deléham, Aug 01 2005
Row sums of number triangle A114608. - Philippe Deléham, Oct 15 2008

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, A lattice on Dyck paths close to the Tamari lattice, arXiv:2309.00426 [math.CO], 2023.
M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.
Marek Bożejko, Maciej Dołęga, Wiktor Ejsmont, and Światosław R. Gal, Reflection length with two parameters in the asymptotic representation theory of type B/C and applications, arXiv:2104.14530 [math.RT], 2021.
F. Chapoton and S. Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv:1310.4521 [math.CO], 2013.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics, Vol. 24, No. 2 (2017), Article P2.3.
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.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 651.
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., Vol. 36, No.4 (2006), pp. 364-387.
Vincent Pilaud and V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016-2017.
Index to sequences related to reversion of series.

Limit of array A102544.

Cf. A000108, A003645, A025225, A052714, A195699.

spec := [S,{B=Union(C,Z),S=Union(B,C),C=Prod(S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

InverseSeries[Series[y-2*y^2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 07 2000 *)
Join[{0},Table[2^n CatalanNumber[n],{n,0,30}]] (* Harvey P. Dale, Aug 29 2015 *)

a(n)=if(n<1,0,2^(n-1)*(2*n-2)!/(n-1)!/n!)

a(n)=if(n<1,0,polcoeff(serreverse(x-2*x^2+x*O(x^n)),n))

a(n)=if(n<1,0,polcoeff(2*x/(1+sqrt(1-8*x+O(x^n))),n))

a(n) = A052714(n)/n!.
a(n) = A003645(n-2)*2, n>1.
a(n) = 8^(n-1)*GAMMA(n-1/2)/GAMMA(n+1)/Pi^(1/2), n>0.
D-finite with recurrence: a(1)=1, (-4+8*n)*a(n) - (n+1)*a(n+1) = 0.
G.f.: (1-sqrt(1-8*x))/4 = x*C(2*x) where C(x) is g.f. for Catalan numbers, A000108.
G.f. A(x) satisfies 2*A(x)^2-A(x)+x=0, A(0)=0 and A(x)=x+2*A(x)^2=x/(1-2*A(x)). Series reversion of x-2*x^2. - Michael Somos, Sep 06 2003
a(0)=0, a(1)=1; a(n) = 2*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
With a different offset, a(0)=1, a(n) = Sum_{k=0..n} Sum_{j=0..n} (j*C(2n-j-1, n-j)*C(j, k)*2^(n-j)/n), n>0. - Paul Barry, Jul 24 2005
The Hankel transform of a(n+1) = [1,2,8,40,224,1344,...] is 4^C(n+1,2). - Philippe Deléham, Nov 06 2007
G.f.: x + 4*x^2/(G(0)-4*x) where G(k) = k*(8*x+1) + 4*x + 2 - 2*x*(2*k+3)*(2*k+4)/G(k+1) ; (continued fraction ). - Sergei N. Gladkovskii, Apr 05 2013
a(n) ~ 8^(n-1)/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Dec 04 2016
From Amiram Eldar, Mar 06 2022: (Start)
Sum_{n>=1} 1/a(n) = 68/49 + 96*arcsin(sqrt(1/8))/(49*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 20/27 - 16*log(2)/81. (End)
a(n) = A025225(n)/2 for n>=1. - Alois P. Heinz, Feb 16 2025

Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Mar 19 2001

Additional comments from Michael Somos, Feb 24 2002

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

A052712 Expansion of e.g.f. (1 + 4x - sqrt(1-8x))/8.

Original entry on oeis.org

0, 1, 2, 24, 480, 13440, 483840, 21288960, 1107025920, 66421555200, 4516665753600, 343266597273600, 28834394170982400, 2652764263730380800, 265276426373038080000, 28649854048288112640000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Also the number of random walk labelings of the 2 X (n-1) king's graph, for n > 1. - Sela Fried, Apr 14 2023

G. C. Greubel, Table of n, a(n) for n = 0..330
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 668
Sela Fried and Toufik Mansour, Graph labelings obtainable by random walks, arXiv:2304.05728 [math.CO], 2023.

Cf. A052711, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A003645.

spec := [S,{B=Prod(C,C),C=Union(B,S),S=Union(B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[n!*2^(n-2)*CatalanNumber[n-1] +Boole[n==1]/2 +Boole[n==0]/4, {n,0,30}] (* G. C. Greubel, May 30 2022 *)

[2^(n-2)*factorial(n)*catalan_number(n-1) +bool(n==0)/8 +bool(n==1)/2 for n in (0..30)] # G. C. Greubel, May 30 2022

D-finite with recurrence: a(0) = 0, a(1)=1, a(2)=2, a(n+1) = 4*(2*n-1)*a(n).
a(n) = 8^(n+1)*Gamma(n+3/2)/sqrt(Pi).
a(n) = n!*A003645(n-2), n>1. - R. J. Mathar, Oct 18 2013
G.f.: (1 + 4*x - 2F0([1,-1/2], [], 8*x))/8. - R. J. Mathar, Jan 25 2020

A107267 A square array of Motzkin related transforms, read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 3, 1, 0, 9, 20, 12, 4, 1, 0, 21, 72, 54, 20, 5, 1, 0, 51, 272, 261, 112, 30, 6, 1, 0, 127, 1064, 1323, 672, 200, 42, 7, 1, 0, 323, 4272, 6939, 4224, 1425, 324, 56, 8, 1, 0, 835, 17504, 37341, 27456, 10625, 2664, 490, 72, 9, 1

Offset: 0

Paul Barry, May 15 2005

Rows are transforms of k^n, k>=0, under the matrix A107131. As a number triangle, with T(n,k)=if(k<=n,sum{j=0..n-k, (1/(j+1))C(j+1,n-k-j+1)C(n-k,j)k^j},0), row sums are A107268 and diagonal sums are A107269. Rows are series reversions of x/(1+kx+kx^2), k>=0. Conjecture: rows count weighted Motzkin paths.

Row k counts colored Motzkin paths, where H(1,0) and U(1,1) each have k colors and D(1,-1) one color. - Paul Barry, May 16 2005

			Array begins
  1, 0,  0,   0,    0,     0,      0, ...
  1, 1,  2,   4,    9,    21,     51, ...
  1, 2,  6,  20,   72,   272,   1064, ...
  1, 3, 12,  54,  261,  1323,   6939, ...
  1, 4, 20, 112,  672,  4224,  27456, ...
  1, 5, 30, 200, 1425, 10625,  81875, ...
  1, 6, 42, 324, 2664, 22896, 203256, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Rows n=0..6 give A000007, A001006, A071356, A107264, A003645, A107265, A107266.
Main diagonal gives A292716.
Cf. A000108.

Number array T(n,k) = Sum_{j=0..k} n^j * binomial(k,j) * binomial(j+1,k-j+1)/(j+1).
G.f. of row k: 1/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017
From Seiichi Manyama, May 05 2019: (Start)
T(n,k) = Sum_{j=0..floor(k/2)} n^(k-j) * binomial(k,2*j) * binomial(2*j,j)/(j+1) = Sum_{j=0..floor(k/2)} n^(k-j) * binomial(k,2*j) * A000108(j).
(k+2) * T(n,k) = n * (2*k+1) * T(n,k-1) - n * (n-4) * (k-1) * T(n,k-2). (End)

A073165 Triangle T(n,k) read by rows: related to David G. Cantor's sigma function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 8, 1, 1, 5, 20, 35, 16, 1, 1, 6, 35, 112, 126, 32, 1, 1, 7, 56, 294, 672, 462, 64, 1, 1, 8, 84, 672, 2772, 4224, 1716, 128, 1, 1, 9, 120, 1386, 9504, 28314, 27456, 6435, 256, 1, 1, 10, 165, 2640, 28314, 151008, 306735, 183040, 24310, 512, 1

Offset: 0

Michael Somos, Jul 24 2002

Square array T(n+k,k) read by antidiagonals: number of stars of length k with n branches.

Row n of T(n+k,k) has g.f. (floor(n/2)+1)F(floor(n/2))(1,3/2,5/2,...,(2*floor(n/2)+1)/2;n,n-1,...,n-floor(n/2)+1;2^n*x) (conjecture). [Paul Barry, Jan 23 2009]

			Triangle rows:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  4,   1;
  1, 4, 10,   8,    1;
  1, 5, 20,  35,   16,    1;
  1, 6, 35, 112,  126,   32,    1;
  1, 7, 56, 294,  672,  462,   64,   1;
  1, 8, 84, 672, 2772, 4224, 1716, 128, 1;

Seiichi Manyama, Rows n = 0..139, flattened
D. G. Cantor, On the analogue of the division polynomials for hyperelliptic curves, J. Reine Angew. Math. (Crelle's J.) 447 (1994), pp. 91-145.
C. Krattenthaler, A. J. Guttmann and X. G. Viennot, Vicious walkers, friendly walkers and Young tableaux, II: with a wall, arXiv:cond-mat/0006367 [cond-mat.stat-mech], 2000.

Square array has main diagonal A049505, columns include A001700, A003645, A000356.

Cf. A133112.

t[n_, k_] := Product[ (n-k+i+j-1) / (i+j-1), {j, 1, k}, {i, 1, j}]; Flatten[ Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, May 23 2012, after PARI *)

{T(n, k) = if( k<0 || k>n, 0, prod( i=1, (k+1)\2, binomial(n + 2*i - 1 - k%2, 4*i - 1 - k%2*2)) / prod( i=0, (k-1)\2, binomial(2*k - 2*i - 1, 2*i)))}

{T(n, k) = if( k<0 || n<0, 0, prod( j=1, k, prod( i=1, j, (n - k + i + j - 1) / (i + j - 1) )))} /* Michael Somos, Oct 16 2006 */

T(n, k) * T(n-2, k-1) - 2 * T(n-1, k-1) * T(n-1, k) + T(n, k-1) * T(n-2, k) = 0.

T(n+k, k) = Product_{1<=i<=j<=k} (n+i+j-1)/(i+j-1). - Ralf Stephan, Mar 02 2005

Edited by Ralf Stephan, Mar 02 2005

A102539 Square array T(n,k) read by antidiagonals: T(n,k) = Product_{1<=i<=j<=k} (n+i+j-1)/(i+j-1).

Original entry on oeis.org

2, 3, 4, 4, 10, 8, 5, 20, 35, 16, 6, 35, 112, 126, 32, 7, 56, 294, 672, 462, 64, 8, 84, 672, 2772, 4224, 1716, 128, 9, 120, 1386, 9504, 28314, 27456, 6435, 256, 10, 165, 2640, 28314, 151008, 306735, 183040, 24310, 512, 11, 220, 4719, 75504, 674817

Offset: 1

Ralf Stephan, Jan 14 2005

Number of semistandard Young tableaux with at most n columns and with entries in [k].

T(n,k) is the number of k X k symmetric matrices with entries in 0..n with each row (and column) in nondecreasing order. - R. H. Hardin, Jul 08 2008

			Square array T(n,k) begins:
  2,  4,    8,    16,     32,       64, ...
  3, 10,   35,   126,    462,     1716, ...
  4, 20,  112,   672,   4224,    27456, ...
  5, 35,  294,  2772,  28314,   306735, ...
  6, 56,  672,  9504, 151008,  2617472, ...
  7, 84, 1386, 28314, 674817, 18076916, ...
  ...

M. Lederer, A determinant-like formula for the Kostka numbers

Rows include A000079, A001700, A003645, A000356.

Main diagonal is A049505.

T[n_, k_] := Product[(n + i + j - 1)/(i + j - 1), {i, 1, k}, {j, i, k}];
Table[T[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 06 2018 *)

It appears that T is identical to the reflected triangle A073165, i.e. T(n, k) = Prod[i=1..floor((k+1)/2), C(n+k+2i-1-(k mod 2), 4i-1-2(k mod 2))] / Prod[i=0..floor((k-1)/2), C(2k-2i-1, 2i)].

A103940 Number of unrooted bipartite n-edge maps in the plane (planar with a distinguished outside face).

Original entry on oeis.org

1, 1, 2, 5, 18, 72, 368, 1982, 11514, 69270, 430384, 2736894, 17752884, 117039548, 782480424, 5294705752, 36206357114, 249894328848, 1739030128872, 12191512867814, 86037243899240, 610827161152012, 4360291880624504, 31280354620428378, 225427088761560916, 1631398499577667252

Offset: 0

Valery A. Liskovets, Mar 17 2005

Bipartite planar maps are dual to Eulerian planar maps.

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Andrew Howroyd, Table of n, a(n) for n = 0..500
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Cf. A003645, A103939, A069727.

a[n_] := (1/(2 n)) (2^(n - 1) Binomial[2 n, n]/(n+1) + Sum[Boole[0 < k < n] EulerPhi[n/k] d[n/k] 2^(k-1) Binomial[2k, k], {k, Divisors[n]}]) + q[n];
d[n_] := If[EvenQ[n], 2, 1];
q[n_] := If[EvenQ[n], 0, 2^((n-1)/2) Binomial[n-1, (n-1)/2]/(n+1)];
Array[a, 25] (* Jean-François Alcover, Aug 30 2019 *)

a(n)={if(n==0, 1, sumdiv(n, d, if(dAndrew Howroyd, Mar 29 2021

For n > 0, a(n) = (1/(2n))*[2^(n-1)*binomial(2n, n)/(n+1) + Sum_{0A000010, d(n)=2, q(n)=0 if n is even and d(n)=1, q(n)=2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1) if n is odd.

More terms from Jean-François Alcover, Aug 30 2019

a(0)=1 prepended by Andrew Howroyd, Mar 29 2021

A240721 Expansion of -(4x + sqrt(1-8x) - 1)/(sqrt(1-8x)(4x^2+x) + 8x^2 - x).

Original entry on oeis.org

1, 7, 49, 351, 2561, 18943, 141569, 1066495, 8085505, 61616127, 471556097, 3621830655, 27902803969, 215530668031, 1668644405249, 12944666918911, 100598145875969, 783027553697791, 6103529011806209, 47636654222999551, 372225072921837569, 2911581699143892991

Offset: 0

Vladimir Kruchinin, Apr 11 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO], 2023.

Cf. A178792.

a := n -> binomial(2*n+2,n)*hypergeom([-n, n+2], [n+3],-1);
seq(round(evalf(a(n), 32)), n=0..19); # Peter Luschny, Jul 16 2014

CoefficientList[Series[-(4 x + Sqrt[1 - 8 x] - 1)/(Sqrt[1 - 8 x] (4 x^2 + x) + 8 x^2 - x), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 12 2014 *)

a(n) := sum((k+1)*binomial(2*(n+1), n-k)*binomial(n+k+1,n), k, 0, n)/(n+1);

a[0]:1$ a[1]:7$ a[2]:49$ a[n] := 8*sum(a[k]*a[n-3-k], k, 0, n-3)+7*sum(a[k]*a[n-2-k], k, 0, n-2)-sum(a[k]*a[n-1-k], k, 0, n-1)+8*a[n-1]$ makelist(a[n], n, 0, 1000); /* Tani Akinari, Jul 16 2014 */

x='x+O('x^50); Vec(-(4*x+sqrt(1-8*x)-1)/(sqrt(1-8*x)*(4*x^2+x)+8*x^2-x)) \\ G. C. Greubel, Apr 05 2017

a(n) = (Sum_{k=0..n} (k+1)*binomial(2*(n+1),n-k)*binomial(n+k+1,n))/(n+1).
a(n) = Sum_{k=0..n} binomial(2*(n+1),k)*2^k*(-1)^(n+k) = binomial(2*(n+1),n+1)*(n+1)*Sum_{k=0..n} binomial(n,k)/(n+k+2). - Max Alekseyev, Jun 16 2021
A(x) = (x*B'(x)+B(x))/(x*B(x)+1) where B(x) = (1-4*x-sqrt(1-8*x))/(8*x^2) is the g.f. of A003645.
a(n) ~ 2^(3*n+3)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 12 2014
a(n) = C(2*n+2, n)*2F1([-n, n+2], [n+3], -1), 2F1 is the hypergeometric function. - Peter Luschny, Jul 16 2014
a(n) = 8*Sum_{k=0..n-3} a(k)*a(n-3-k) + 7*Sum_{k=0..n-2} a(k)*a(n-2-k) - Sum_{k=0..n-1} a(k)*a(n-1-k) + 8*a(n-1) for n > 2, a(0)=1, a(1)=7, a(2)=49. - Tani Akinari, Jul 16 2014
D-finite with recurrence -(n+1)*(3*n-2)*a(n) +(21*n^2-5*n-2)*a(n-1) +4*(3*n+1)*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
a(n) = 2^(n+1)*binomial(2*n+1,n) - A178792(n). - Akiva Weinberger, Dec 04 2024

A269796 a(n) = 4^n * A000108(n+1).

Original entry on oeis.org

1, 8, 80, 896, 10752, 135168, 1757184, 23429120, 318636032, 4402970624, 61641588736, 872465563648, 12463793766400, 179478630236160, 2602440138424320, 37965009078190080, 556820133146787840, 8205770383215820800, 121445401671594147840, 1804331681977970196480, 26900945076762464747520

Offset: 0

Michel Marcus, Mar 07 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016. Cf. 2.1.10.

Cf. A000108, A003645, A101600, A098430.

[4^n*Catalan(n+1): n in [0..25]]; // Vincenzo Librandi, Apr 25 2016

Table[4^n CatalanNumber[n + 1], {n, 0, 20}] (* Bruno Berselli, Mar 07 2016 *)

cat(n) = binomial(2*n,n)/(n+1);
a(n) = 4^n*cat(n+1);

[4^n*catalan_number(n+1) for n in (0..20)] # Bruno Berselli, Mar 07 2016

G.f.: (1 - sqrt(1 - 16*x) - 8*x)/(32*x^2). - Bruno Berselli, Mar 07 2016
a(n) = -2^(4*n + 3)*binomial(n + 1/2, -3/2), after Peter Luschny in A000108. - Bruno Berselli, Mar 07 2016
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 52/75 + 512*arcsin(1/4)/(75*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 164/289 + 1536*arcsinh(1/4)/(289*sqrt(17)). (End)
From Karol A. Penson, Aug 26 2025: (Start)
O.g.f.: exp(Sum_{n>=1} A098430(n)*x^n/n).
O.g.f.: exp(Sum_{n>=1} 4^n*(2*n)!*x^n/(n*(n!)^2)).
a(n) = 4^n*binomial(2*n + 2, n + 1)/(n + 2).
a(n) = Integral_{x=0..16} (x^n*sqrt(x)*sqrt(1 - x/16)/(8*Pi)) dx. (End)

cp's OEIS Frontend

A201639 Triangle read by rows, T(n,k) for 0<=k<=n, generalizes the Motzkin lattice paths with weights of A003645.

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A052701 a(0) = 0; for n >= 1, a(n) = 2^(n-1)*C(n-1), where C(n) = A000108(n) Catalan numbers, n>0.

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

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A052712 Expansion of e.g.f. (1 + 4*x - sqrt(1-8*x))/8.

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A107267 A square array of Motzkin related transforms, read by antidiagonals.

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A073165 Triangle T(n,k) read by rows: related to David G. Cantor's sigma function.

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A102539 Square array T(n,k) read by antidiagonals: T(n,k) = Product_{1<=i<=j<=k} (n+i+j-1)/(i+j-1).

A052712 Expansion of e.g.f. (1 + 4x - sqrt(1-8x))/8.

A240721 Expansion of -(4x + sqrt(1-8x) - 1)/(sqrt(1-8x)(4x^2+x) + 8x^2 - x).