A052701 -id:A052701 - cp's OEIS frontend

A064306 Convolution of A052701 (Catalan numbers multiplied by powers of 2) with powers of -1.

1, 1, 7, 33, 191, 1153, 7295, 47617, 318463, 2170881, 15028223, 105365505, 746651647, 5339185153, 38478839807, 279201841153, 2037998419967, 14954803494913, 110255315877887, 816299567480833, 6066679566041087

Offset: 0

Wolfdieter Lang, Sep 13 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
W. Lang, On polynomials related to derivatives of the generating function of Catalan numbers, Fib. Quart. 40,4 (2002) 299-313; Eq.(31) with lambda=-1/2.

CoefficientList[Series[(1-Sqrt[1-8*x])/(4*x*(1+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 09 2013 *)
Table[FullSimplify[2^(n+1)*(2*n+2)! * Hypergeometric2F1Regularized[1, n+3/2, n+3, -8]/(n+1)! + (-1)^n/2],{n,0,20}] (* Vaclav Kotesovec, Dec 09 2013 *)
Table[(-1)^n*Sum[(-2)^k * CatalanNumber[k], {k,0,n}], {n,0,50}] (* G. C. Greubel, Jan 27 2017 *)

for(n=0, 25, print1((-1)^n*sum(k=0,n, (-2)^k*binomial(2*k,k)/(k+1)), ", ")) \\ G. C. Greubel, Jan 27 2017

def A064306():
    f, c, n = 1, 1, 1
    while True:
        yield f
        n += 1
        c = c * (8*n - 12) // n
        f = c - f
a = A064306()
print([next(a) for  in range(21)]) # _Peter Luschny, Nov 30 2016

a(n) = (-1)^n*Sum_{k=0,..,n} (C(k)/(-1/2)^k) with C(k)=A000108(k) (Catalan).
a(n) = -a(n-1) + C(n)*2^n, n >= 0, a(-1) := 0, with C(n)=A000108(n).
G.f.: A(2*x)/(1+x), with A(x) g.f. of Catalan numbers A000108.
Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Dec 09 2013
a(n) ~ 2^(3*n+3)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 09 2013

A151374 Number of walks within N^2 (the first quadrant of Z^2) starting at (0, 0), ending on the vertical axis and consisting of 2n steps taken from {(-1, -1), (-1, 0), (1, 1)}.

Original entry on oeis.org

1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, 17199104, 120393728, 852017152, 6085836800, 43818024960, 317680680960, 2317200261120, 16992801914880, 125210119372800, 926554883358720, 6882979133521920, 51309480813527040, 383705682605506560, 2877792619541299200

Offset: 0

Manuel Kauers, Nov 18 2008

A052701 shifted one place left. - R. J. Mathar, Dec 13 2008
Expansion of c(2*x), where c(x) is the g.f. of A000108. - Philippe Deléham, Feb 26 2009; simplified by Alexander Burstein, Jul 31 2018
From Joerg Arndt, Oct 22 2012: (Start)
Also the number of strings of length 2*n of two different types of balanced parentheses.
For example, a(1) = 2, since the two possible strings of length 2 are [] and (), a(2) = 8, since the 8 possible strings of length 4 are (()), [()], ([]), [[]], ()(), [](), ()[], and [][].
The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)
Number of Dyck paths of length 2n in which the step U=(1,1) come in 2 colors. - José Luis Ramírez Ramírez, Jan 31 2013
Row sums of triangle in A085880. - Philippe Deléham, Nov 15 2013
Hankel transform is 2^(n+n^2) = A053763(n+1). - Philippe Deléham, Nov 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
J. Bouttier, P. Di Francesco and E. Guitter, Statistics of planar graphs viewed from a vertex: a study via labeled trees, Nucl. Phys. B, Vol. 675, No. 3 (2003), pp. 631-660. See p. 631, eq. (3.3).
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.
Vedran Čačić and Vjekoslav Kovač, On the fraction of IL formulas that have normal forms, arXiv:1309.3408 [math.LO], 2013.
Stefano Capparelli and Alberto Del Fra, Dyck Paths, Motzkin Paths, and the Binomial Transform, Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.5.
Grégory Chatel and Vincent Pilaud, Cambrian Hopf algebras, Adv. Math. 311, 598-633 (2017). Prop. 3.
Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schröder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016. See eq. (1.13), a=b=2.
Nicolas Crampe, Julien Gaboriaud and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.
Hsien-Kuei Hwang, Mihyun Kang and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
Bradley Robert Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Georg Muntingh, Implicit Divided Differences, Little Schröder Numbers and Catalan Numbers, J. Int. Seq., Vol. 15 (2012), Article 12.6.5; arXiv preprint, arXiv:1204.2709 [math.CO], 2012.
L. Poulain d'Andecy, Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics, arXiv:2304.00850 [math.RT], 2023. See p. 64.

Cf. A000108, A046521, A052701, A053763, A085880.

[2^n * Catalan(n): n in [0..25]]; // Vincenzo Librandi, Oct 24 2012

A151374_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 2*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od;
convert(a,list)end: A151374_list(23); # Peter Luschny, May 19 2011

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]

my(x='x+O('x^66)); Vec(sqrt(2-8*x-2*sqrt(1-8*x))/(4*x)) \\ Joerg Arndt, May 11 2013

def A151374():
    a, n = 1, 1
    while True:
        yield a
        n += 1
        a = a * (8*n - 12) // n
A = A151374()
print([next(A) for  in range(24)]) # _Peter Luschny, Nov 30 2016

a(n) = 2^n * A000108(n). - Philippe Deléham, Feb 01 2009
From Gary W. Adamson, Jul 12 2011: (Start)
a(n) is the top left term in M^n, M = the following infinite square production matrix:
   2, 2, 0, 0, 0, 0, ...
   2, 2, 2, 0, 0, 0, ...
   2, 2, 2, 2, 0, 0, ...
   2, 2, 2, 2, 2, 0, ...
   2, 2, 2, 2, 2, 2, ...
   ...
(End)
E.g.f.: KummerM(1/2, 2, 8*x). - Peter Luschny, Aug 26 2012
From Sergei N. Gladkovskii, Apr 05 2013: (Start)
E.g.f.: Let F(x)=Sum_{n>=0} a(n)*x^n/(2*n)!, then F(x) = E(0)/(1-sqrt(x)) where E(k) = 1 - sqrt(x)/(1 - sqrt(x)/(sqrt(x) - (k+1)*(k+2)/2/E(k+1) )); (continued fraction ).
G.f.: 1 + 4*x/(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). (End)
G.f.: sqrt(2-8*x-2*sqrt(1-8*x))/(4*x). - Mark van Hoeij, May 10 2013
G.f.: (1-sqrt(1-8*x))/(4*x). - Philippe Deléham, Nov 15 2013
D-finite with recurrence (n+1)*a(n) + 4*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Mar 05 2014
a(n) = 4^n*2F1((1-n)/2,-n/2;1;1)/(n+1). - Benedict W. J. Irwin, Jul 12 2016
a(n) ~ 8^n*n^(-3/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016
From Peter Bala, Aug 17 2021: (Start)
a(n) = Sum_{k = 0..floor(n/2)} A046521(n,2*k)*Catalan(2*k).
G.f.: A(x) = 1/sqrt(1 - 4*x)*e(x/(1 - 4*x)), where e(x) = (c(x) + c(-x))/2 is the even part of the function c(x) = (1 - sqrt(1 - 4*x))/(2*x), the g.f. of the Catalan numbers A000108. Inversely, (c(x) + c(-x))/2 = 1/sqrt(1 + 4*x)*A(x/(1 + 4*x)).
x*A(x) = Series reversion of (x - 2*x^2). (End)
Sum_{n>=0} 1/a(n) = 68/49 + 96*arctan(1/sqrt(7)) / (49*sqrt(7)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 20/27 - 16*log(2)/81. - Amiram Eldar, Jan 25 2022
G.f.: 1/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-...))))))))) (continued fraction). - Nikolaos Pantelidis, Nov 20 2022
a(n) = 2*Sum_{k=1..n} a(k-1)*a(n-k), a(0) = 1. - Mehdi Naima, Jan 16 2023

A052714 a(n) = 2^(n-1) * n! * Catalan(n-1) for n > 0 with a(0) = 0.

Original entry on oeis.org

0, 1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400, 9033331507200, 686533194547200, 57668788341964800, 5305528527460761600, 530552852746076160000, 57299708096576225280000, 6646766139202842132480000, 824199001261152424427520000

Offset: 0

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

a(n+1) is the number of square roots of any permutation in S_{8*n} whose disjoint cycle decomposition consists of 2*n cycles of length 4. - Luis Manuel Rivera Martínez, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 670.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), this sequence (or A144828) (m=2), A221954 (m=3), A052734 (m=4), A221953 (m=5), A221955 (m=6).

Cf. A000108, A221953, A221954.

[0] cat [Catalan(n-1)*2^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

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

Join[{0}, Table[CatalanNumber[n-1] 2^(n-1) n!, {n,  1, 20}]] (* Vincenzo Librandi, Mar 11 2013 *)

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

[0]+[2^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

E.g.f.: (1- sqrt(1-8*x))/4.
Recurrence: a(1) = 1, 4*(1 - 2*n)*a(n) + a(n+1) = 0.
a(n) = A052701(n)*n!.
a(n) = 8^(n-1)*Gamma(n-1/2)/Pi^(1/2), n>0.
a(n+1) = A090802(2n, n). - Ross La Haye, Oct 18 2005
a(n) = 2^(n-1)*(2*n-2)!/(n-1)! for n>=1.
E.g.f. A(x) satisfies differential equation A'(x)=1/(1-4*A(x)). - Vladimir Kruchinin, May 04 2011
G.f.: x/(1-4x/(1-8x/(1-12x/(1-16x/(1-20x/(1-24x/(1-28x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
G.f.: 2*x/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+4)/(2*x*(8*k+4) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(0) = 0, a(1) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 09 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/8)*sqrt(Pi)*erf(1/(2*sqrt(2)))/(2*sqrt(2)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/8)*sqrt(Pi)*erfi(1/(2*sqrt(2)))/(2*sqrt(2)), where erfi is the imaginary error function. (End)

Edited by N. J. A. Sloane, Feb 03 2013

A003645 a(n) = 2^n * C(n+1), where C(n) = A000108(n) Catalan numbers.

Original entry on oeis.org

1, 4, 20, 112, 672, 4224, 27456, 183040, 1244672, 8599552, 60196864, 426008576, 3042918400, 21909012480, 158840340480, 1158600130560, 8496400957440, 62605059686400, 463277441679360, 3441489566760960, 25654740406763520, 191852841302753280, 1438896309770649600

Offset: 0

N. J. A. Sloane

Number of nonisomorphic unrooted unicursal planar maps with n+2 edges and exactly one vertex of valency 1 (unicursal means that exactly two vertices are of odd valency). - Valery A. Liskovets, Apr 07 2002
Total number of vertices in rooted Eulerian planar maps with n+1 edges.
Half the number of ways to dog-ear every page of an (n+1)-page book. - R. H. Hardin, Jun 21 2002
Convolution of A052701(n+1) with itself.
Number of Motzkin lattice paths with weights: 1 for up step, 4 for level step and 4 for down step. - Wenjin Woan, Oct 24 2004
The number of rooted bipartite n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
Also the number of paths of length 2n+1 in a binary tree between two vertices that are one step away from each other. - David Koslicki (koslicki(AT)math.psu.edu), Nov 02 2010
2*a(n) for n > 1 is the number of increasing strict binary trees with 2n-1 nodes that simultaneously avoid 213 and 231 in the classical sense. For more information about increasing strict binary trees with an associated permutation, see A245894. - Manda Riehl, Aug 22 2014

L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin. 54 (2000), 149-160.
V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

A. Claesson, S. Kitaev and A. de Mier, An involution on bicubic maps and beta(0,1)-trees, arXiv preprint arXiv:1210.3219 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 01 2013
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, 2017.
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 652.
Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 13.
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., Vol. 282, No. 1-3 (2004), pp. 209-221.
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.
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
Youngja Park and SeungKyung Park, Enumeration of generalized lattice paths by string types, peaks, and ascents, Discrete Mathematics, Vol. 339, No. 11 (2016), pp. 2652-2659.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seq., Vol. 9 (2006), Article 06.1.1.

Cf. A069724, A069725, A090375.
Third row of array A102539.
Column of array A073165.
Cf. A000108, A052701, A195699.

[2^n*Binomial(2*n+3, n+1)/(2*n+3) : n in [0..30]]; // Wesley Ivan Hurt, Aug 23 2014

A003645:=n->2^n*binomial(2*n+3, n+1)/(2*n+3): seq(A003645(n), n=0..30); # Wesley Ivan Hurt, Aug 23 2014

Table[2^n CatalanNumber[n+1],{n,0,20}] (* Harvey P. Dale, May 07 2013 *)

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

a(n) = A052701(n+2)/2.
2*a(n) matches the odd-indexed terms of A090375.
a(n) = 2^n * binomial(2n+3, n+1) / (2n+3). - Len Smiley, Feb 24 2006
G.f.: (1-4x-sqrt(1-8x))/(8x^2) = C(2x)^2, where C(x) is the g.f. for Catalan numbers, A000108.
From Gary W. Adamson, Jul 12 2011: (Start)
Let M = the following production matrix:
  2, 2, 0, 0, 0, ...
  2, 2, 2, 0, 0, ...
  2, 2, 2, 2, 0, ...
  2, 2, 2, 2, 2, ...
  ...
a(n) = sum of top row terms in M^n. Example: top row of M^3 = (40, 40, 24, 8, 0, 0, 0, ...), sum = 112 = a(3). (End)
D-finite with recurrence (n+2)*a(n) - 4*(2n+1)*a(n-1) = 0. - R. J. Mathar, Apr 01 2012
E.g.f.: a(n) = n!* [x^n] exp(4*x)*BesselI(1, 4*x)/(2*x). - Peter Luschny, Aug 25 2012
Expansion of square of continued fraction 1/(1 - 2*x/(1 - 2*x/(1 - 2*x/(1 - ...)))). - Ilya Gutkovskiy, Apr 19 2017
From Amiram Eldar, Mar 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 38/49 + 192*arcsin(sqrt(1/8))/(49*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 14/27 + 32*log(2)/81. (End)
a(n) = Product_{1 <= i <= j <= n} (i + j + 2)/(i + j - 1). Cf. A001700. - Peter Bala, Feb 22 2023

A025225 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)a(1) for n >= 2. Also a(n) = (2^n)C(n-1), where C = A000108 (Catalan numbers).

Original entry on oeis.org

2, 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688, 34398208, 240787456, 1704034304, 12171673600, 87636049920, 635361361920, 4634400522240, 33985603829760, 250420238745600, 1853109766717440, 13765958267043840, 102618961627054080, 767411365211013120

Offset: 1

Clark Kimberling

Number of generators of degree n of the Hopf algebra of 2-colored planar binary trees. Also, dimensions of the graded components of the primitive Lie algebra of the same Hopf algebra. - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

Alois P. Heinz, Table of n, a(n) for n = 1..350
Suzanne Bobzien, The combinatorics of the Stoic conjunction: Hipparchus refuted and Chrysippus vindicated, Oxford Studies in Ancient Philosophy, Vol. XL, Summer 2011, pp. 157-188.
L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337, remark 3.3.
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 653.
J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008.
Volkan Yildiz, Counting false entries in truth tables of bracketed formulas connected by m-implication, arXiv:1203.4645 [math.CO], 2012.
Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv:1205.5595 [math.CO], 2012.

Essentially identical to A115125.

Cf. A052701.

[2^n*Catalan(n-1): n in [1..30]]; // Vincenzo Librandi, Nov 06 2016

a:= n-> (2^n)*binomial(2*n-2, n-1)/n:
seq(a(n), n=1..25); # Alois P. Heinz, Jan 27 2012

InverseSeries[Series[y/2-y^2/2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 13 2000 *)
a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Jul 09 2013 *)

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

G.f.: (1-sqrt(1-8*x))/2. - Michael Somos, Jun 08 2000
Given g.f. C(x) and given A(x)= g.f. of A100238, then B(x)=A(x)-1-x satisfies B(x)=x-C(x*B(x)). - Michael Somos, Sep 07 2005
n*a(n) + 4*(-2*n+3)*a(n-1) = 0. - R. J. Mathar, Feb 25 2015
a(n) = 2*A052701(n). - Alois P. Heinz, Feb 16 2025

Typo in definition corrected by R. J. Mathar, Aug 11 2008

A110506 Riordan array (1/(1-xc(2x)),xc(2x)/(1-xc(2x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 13, 19, 7, 1, 67, 102, 44, 10, 1, 381, 593, 278, 78, 13, 1, 2307, 3640, 1795, 568, 121, 16, 1, 14589, 23231, 11849, 4051, 999, 173, 19, 1, 95235, 152650, 79750, 28770, 7820, 1598, 234, 22, 1, 636925, 1025965, 545680, 204760, 59650, 13642, 2392, 304, 25, 1

Offset: 0

Paul Barry, Jul 24 2005

Deleham triangle Delta(0^n,2-0^n) [see construction in A084938]. The binomial transform of the inverse of this triangle has general element (-2)^(n-k)*C(k,n-k), that is, it is the Riordan array (1,x(1-2x)) [A110509]. Row sums are A052701. Diagonal sums are A110508. Inverse is A110511.

			Rows begin:
1;
1,1;
3,4,1;
13,19,7,1;
67,102,44,10,1;
381,593,278,78,13,1;
From _Philippe Deléham_, Dec 01 2015: (Start)
Production matrix begins:
1, 1
2, 3, 1
2, 4, 3, 1
2, 4, 4, 3, 1
2, 4, 4, 4, 3, 1
2, 4, 4, 4, 4, 3, 1
2, 4, 4, 4, 4, 4, 3, 1
(End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000108, A052701, A084938, A110508, A110509, A110511, A114189.

{{1}}~Join~Table[Sum[j Binomial[2 n - j - 1, n - j] Binomial[j, k] 2^(n - j), {j, 0, n}]/n, {n, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)

tabl(nn)= {for (n=0, nn, for (k=0, n, if (n==0, x = 0^k, x = sum(j=0, n, j*binomial(2*n-j-1, n-j)*binomial(j, k)*2^(n-j)/n)); print1(x, ", ");); print(););} \\ Michel Marcus, Jun 18 2015

T(0,0) = 1, T(n,k) = (Sum_{j=0..n} j*C(2*n-j-1,n-j) * C(j,k) * 2^(n-j))/n.

T(n,k) = (-1)^(n-k)*A114189(n,k). - Philippe Deléham, Mar 24 2007

A354733 a(0) = a(1) = 1; a(n) = 2 * Sum_{k=0..n-2} a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 2, 4, 10, 24, 64, 168, 464, 1280, 3624, 10304, 29728, 86240, 252480, 743040, 2200640, 6547200, 19571200, 58727680, 176883200, 534476800, 1619912320, 4923070464, 14999764480, 45807916544, 140196076544, 429931051008, 1320905583616, 4065358827520

Offset: 0

Ilya Gutkovskiy, Jun 04 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000108, A007477, A052701, A151374, A354734, A354735, A354736.

a[0] = a[1] = 1; a[n_] := a[n] = 2 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 29}]
nmax = 29; CoefficientList[Series[(1 - Sqrt[1 - 8 x^2 (1 + x)])/(4 x^2), {x, 0, nmax}], x]

a(n) = sum(k=0, n\2, 2^k*binomial(k+1, n-2*k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Nov 05 2023

G.f. A(x) satisfies: A(x) = 1 + x + 2 * (x * A(x))^2.
G.f.: (1 - sqrt(1 - 8 * x^2 * (1 + x))) / (4 * x^2).
a(n) ~ 5^(1/4) * (1 + sqrt(5))^(n+2) / (8 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 04 2022
a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(k+1,n-2*k) * A000108(k). - Seiichi Manyama, Nov 05 2023

A337168 a(n) = (-1)^n + 2 * Sum_{k=0..n-1} a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 5, 21, 105, 553, 3053, 17405, 101713, 606033, 3667797, 22485477, 139340985, 871429497, 5492959293, 34862161869, 222592918689, 1428814897825, 9215016141989, 59684122637237, 388045493943049, 2531696701375689, 16569559364596365, 108758426952823709

Offset: 0

Ilya Gutkovskiy, Jan 28 2021

nonn

Inverse binomial transform of A151374.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See pp. 97-98.

Cf. A000108, A005043, A052701, A151374, A162326, A337169.

a[n_] := a[n] = (-1)^n + 2 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k CatalanNumber[k], {k, 0, n}], {n, 0, 23}]
Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 8], {n, 0, 23}]

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + 2*x*A(x)^2.
G.f.: (1 - sqrt(1 - 8*x / (1 + x))) / (4*x).
E.g.f.: exp(3*x) * (BesselI(0,4*x) - BesselI(1,4*x)).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * Catalan(k).
a(n) ~ 7^(n + 3/2) / (2^(9/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021

A098579 Expansion of sqrt(1-8*x).

Original entry on oeis.org

1, -4, -8, -32, -160, -896, -5376, -33792, -219648, -1464320, -9957376, -68796416, -481574912, -3408068608, -24343347200, -175272099840, -1270722723840, -9268801044480, -67971207659520, -500840477491200, -3706219533434880, -27531916534087680, -205237923254108160

Offset: 0

Paul Barry, Sep 16 2004

If A(x) is the g.f. and B(x) := A(-x), then A(x) * B(x) = A(8*x^2), B(x)^2 - A(x)^2 = 16*x, B(x)^2 + A(x)^2 = 2. Also, if U = x*A(x^2/4), T = x*A(-x^2/4), then (T^2 + U^2)^2 = T^2 - U^2 is a parametrization of the lemniscate of Bernoulli. - Michael Somos, Aug 22 2019

			G.f. = 1 - 4*x - 8*x^2 - 32*x^3 - 160*x^4 - 896*x^5 - 5376*x^6 + ... - _Michael Somos_, Aug 22 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A052701, A002420.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!(Sqrt(1-8*x))); // G. C. Greubel, Feb 03 2018

CoefficientList[Series[Sqrt[1-8x],{x,0,30}],x] (* or *) Table[(8^(x-1) Pochhammer[-(1/2),x-1])/Pochhammer[1,x-1],{x,30}] (* Harvey P. Dale, Jan 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -CatalanNumber[n - 1] 2^(n + 1)]; (* Michael Somos, Aug 22 2019 *)

my(x='x+O('x^30)); Vec(sqrt(1-8*x)) \\ G. C. Greubel, Feb 03 2018

G.f.: 4*x*C(2*x)-1, where C(x) is the g.f. for the Catalan numbers A000108; a(0)=1, a(n)=-2^(n+1)*binomial(2(n-1), n-1)/n, n>0.
D-finite with recurrence: n*a(n) +4*(3-2*n)*a(n-1)=0. - R. J. Mathar, Nov 09 2012
0 = a(n)*(+64*a(n+1) -20*a(n+2)) +a(n+1)*(+4*a(n+1) +a(n+2)) for all n in Z. - Michael Somos, Aug 22 2019
a(n) ~ -2^(3*n-1) / (sqrt(Pi) * n^(3/2)). - Amiram Eldar, Sep 01 2025

A103939 Number of unrooted Eulerian n-edge maps in the plane (planar with a distinguished outside face).

Original entry on oeis.org

1, 1, 3, 8, 32, 136, 722, 3924, 22954, 138316, 860364, 5472444, 35503288, 234070648, 1564945158, 10589356592, 72412611194, 499788291616, 3478059566250, 24383023246284, 172074483068320, 1221654305104920, 8720583728414354, 62560709120463028, 450854177292364660

Offset: 0

Valery A. Liskovets, Mar 17 2005

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. A052701, A103940, A069727.

a[n_] := (1/(2n)) (2^n Binomial[2n, n]/(n+1) + Sum[Boole[0Jean-François Alcover, Aug 28 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*binomial(2n, n)/(n+1) + Sum_{0A000010.

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 29 2021

cp's OEIS Frontend

A064306 Convolution of A052701 (Catalan numbers multiplied by powers of 2) with powers of -1.

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A151374 Number of walks within N^2 (the first quadrant of Z^2) starting at (0, 0), ending on the vertical axis and consisting of 2n steps taken from {(-1, -1), (-1, 0), (1, 1)}.

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A052714 a(n) = 2^(n-1) * n! * Catalan(n-1) for n > 0 with a(0) = 0.

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A003645 a(n) = 2^n * C(n+1), where C(n) = A000108(n) Catalan numbers.

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A025225 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also a(n) = (2^n)*C(n-1), where C = A000108 (Catalan numbers).

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A110506 Riordan array (1/(1-xc(2x)),xc(2x)/(1-xc(2x))), c(x) the g.f. of A000108.

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A025225 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)a(1) for n >= 2. Also a(n) = (2^n)C(n-1), where C = A000108 (Catalan numbers).