A025225 -id:A025225 - cp's OEIS frontend

A141318 Euler transform of A025225.

1, 2, 7, 28, 139, 762, 4549, 28464, 184742, 1230348, 8362298, 57756888, 404218046, 2860222020, 20428192842, 147072637472, 1066221149519, 7776835940430, 57028407361537, 420199462347876, 3109418928328473, 23098206138595710, 172184651286785655, 1287629984135461888

Offset: 0

Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

nonn

Dimensions of the graded components of the domain of cocommutativity of the Hopf algebra of 2-colored planar binary trees.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.

Cf. A025225.

with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(add((2^d)*binomial(2*d-2,d-1),
                   d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 27 2012

a[n_] := a[n] = If[n==0, 1, Sum[Sum[2^d*Binomial[2*d-2, d-1], {d, Divisors[ j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

a(n) ~ c * 8^n / n^(3/2), where c = 0.23667440081254669364903300472... . - Vaclav Kotesovec, Sep 04 2014

a(0)=1 prepended by Alois P. Heinz, Jul 28 2017

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

A052713 Expansion of e.g.f. (1-sqrt(1-8*x))/2.

Original entry on oeis.org

0, 2, 8, 96, 1920, 53760, 1935360, 85155840, 4428103680, 265686220800, 18066663014400, 1373066389094400, 115337576683929600, 10611057054921523200, 1061105705492152320000, 114599416193152450560000

Offset: 0

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

Has a square root singularity.

G. C. Greubel, Table of n, a(n) for n = 0..330
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 669

Cf. A052711, A052712, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A025225.

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

Table[n!*2^n*CatalanNumber[n-1] + Boole[n==0], {n,0,30}] (* G. C. Greubel, May 29 2022 *)

[2^n*factorial(n)*catalan_number(n-1) + bool(n==0)/2 for n in (0..30)] # G. C. Greubel, May 29 2022

D-finite with recurrence: a(1)=2, a(n+1) = 4*(2*n -1)*a(n).
a(n+1) = 1/4*8^(n+1)*Gamma(n+1/2)/Pi^(1/2)
a(n+1) = ((2*n)!/n!)*2^(n+1). - Zerinvary Lajos, Sep 25 2006
a(n) = n!*A025225(n). - R. J. Mathar, Oct 18 2013
G.f.: (1- 2F0([1,-1/2], [], 8*x))/2. - R. J. Mathar, Jan 25 2020

A371658 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x))^2.

Original entry on oeis.org

1, 4, 48, 784, 14784, 302976, 6555648, 147380480, 3408817152, 80592320512, 1938923790336, 47314993324032, 1168315059240960, 29136848453632000, 732857340425011200, 18569095605771632640, 473534596510970019840, 12144227894941523116032

Offset: 0

Seiichi Manyama, Apr 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..694

Cf. A219534, A219537, A371657.
Cf. A025225, A371655, A371661.
Cf. A219538.

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 4^(n-k)*binomial(n, k)*binomial(3*n-k, n-1-2*k))/n);

a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 4^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0.

a(n) = 2^n * A219538(n). - Seiichi Manyama, Dec 26 2024

A100225 G.f. A(x) satisfies: 3^n - 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (3+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

Original entry on oeis.org

1, 1, 2, 0, -4, 0, 16, 0, -80, 0, 448, 0, -2688, 0, 16896, 0, -109824, 0, 732160, 0, -4978688, 0, 34398208, 0, -240787456, 0, 1704034304, 0, -12171673600, 0, 87636049920, 0, -635361361920, 0, 4634400522240, 0, -33985603829760, 0, 250420238745600, 0, -1853109766717440

Offset: 0

Paul D. Hanna, Nov 28 2004

sign

More generally, if g.f. A(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]A(x)^n, then A(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z and A(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2.

			From the table of powers of A(x) (A100226), we see that
3^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,1],2,0,-4,0,16,0,-80,...
A^2=[1,2,5],4,-4,-8,16,32,-80,...
A^3=[1,3,9,13],6,-12,-4,48,0,...
A^4=[1,4,14,28,33],8,-24,16,80,...
A^5=[1,5,20,50,85,81],10,-40,60,..
A^6=[1,6,27,80,171,246,197],12,-60,...
the main diagonal of which is A100227 = [1,5,13,33,81,197,477,...], where Sum_{n>=1} (A100227(n)/n)*x^n = log((1-x)/(1-2*x-x^2)).

Cf. A100226, A100227, A000108, A025225, A100223, A100228.

a(n)=if(n==0,1,(3^n-1-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)

a(n)=if(n==0,1,if(n==1,1,if(n==2,2,-8*(n-3)*a(n-2)/n)))

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

G.f.: (1+2*x+sqrt(1+8*x^2))/2. G.f.: A(x) = x/(series_reversion[x*(1-x)/(1-2*x-x^2)]). a(n) = -8*(n-3)*a(n-2)/n for n>2, with a(0)=1, a(1)=1, a(2)=2. a(2*n) = 2^n*(-1)^(n-1)*A000108(n-1), a(2*n+1)=0, for n>=1, where A000108=Catalan.

A100238 G.f. A(x) satisfies: 2^n + 1 = Sum_{k=0..n} [x^k] A(x)^n for n>=1.

Original entry on oeis.org

1, 2, -2, 4, -12, 40, -144, 544, -2128, 8544, -35008, 145792, -615296, 2625792, -11311616, 49124352, -214838528, 945350144, -4182412288, 18593224704, -83015133184, 372090122240, -1673660915712, 7552262979584, -34178799378432, 155096251351040, -705533929816064

Offset: 0

Paul D. Hanna, Nov 30 2004

sign

			From the table of powers of A(x), we see that
2^n+1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1: [1, 2], -2, 4, -12, 40, -144, 544, -2128, 8544, ...;
A^2: [1, 4, 0], 0, -4, 16, -64, 256, -1040, 4288, ...;
A^3: [1, 6, 6, -4], 0, 0, -8, 48, -240, 1120, -5088, ...;
A^4: [1, 8, 16, 0, -8], 0, 0, 0, -16, 128, -768, ...;
A^5: [1, 10, 30, 20, -20, -8], 0, 0, 0, 0, -32, ...;
A^6: [1, 12, 48, 64, -12, -48, 0], 0, 0, 0, 0, 0, ...;
A^7: [1, 14, 70, 140, 56, -112, -56, 16], 0, 0, 0, ...;
A^8: [1, 16, 96, 256, 240, -128, -256, 0, 32], 0, 0, ...; ...
In the above table of coefficients in A(x)^n, the main diagonal satisfies:
[x^n] A(x)^(n+1) = (n+1)*A009545(n+1) for n>=0.

Cf. A100223, A071356, A009545, A143045.

a(n) = -(-1)^n * A025227(n), if n>1.

{a(n)=if(n==0,1,(2^n+1-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)}

{a(n)=if(n==0,1,if(n==1,2,if(n==2,-2,(-2*(2*n-3)*a(n-1)+4*(n-3)*a(n-2))/n)))}

{a(n)=polcoeff( (1+2*x+sqrt(1+4*x-4*x^2+x^2*O(x^n)))/2,n)}

a(n)=polcoeff((1+2*x+sqrt(1+4*x-4*x^2+x*O(x^n)))/2,n)

a(n) = (-2*(2*n-3)*a(n-1) + 4*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=2, a(2)=-2.
G.f.: A(x) = (1+2*x + sqrt(1+4*x-4*x^2))/2.
G.f. satisfies: (2+z)^n + (1+z)^n - z^n = Sum_{k=0..n} [x^k] (A(x)+z*x)^n for all z, where [x^k] F(x) denotes the coefficient of x^k in F(x).
Given g.f. A(x), then B(x)=A(x)-1-x series reversion is -B(-x). - Michael Somos, Sep 07 2005
Given g.f. A(x) and C(x) = g.f. of A025225, then B(x)=A(x)-1-x satisfies B(x)=x-C(x*B(x)). - Michael Somos, Sep 07 2005
G.f.: 4x^2/(1+2x - sqrt(1+4x-4x^2)). - Michael Somos, Sep 08 2005

A115125 A sequence related to Catalan numbers A000108.

Original entry on oeis.org

1, 2, 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688, 34398208, 240787456, 1704034304, 12171673600, 87636049920, 635361361920, 4634400522240, 33985603829760, 250420238745600, 1853109766717440, 13765958267043840, 102618961627054080, 767411365211013120

Offset: 0

Wolfdieter Lang, Jan 13 2006

Essentially identical to A025225.
The convolution of this sequence with the sequence {(-1)^n} is A064062 (see also A062992).
The sequence A064062 appears in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta=1 (or alpha=1, beta=2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.

Cf. A000108, A025225, A062992, A064062, A064094.

[1] cat [2^n*Binomial(2*n-2, n-1)/n: n in [1..30]]; // G. C. Greubel, May 03 2018

a:= n-> `if`(n=0, 1, 2^n*binomial(2*n-2, n-1)/n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2022

a[0] = 1; a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 09 2013 *)

a(n)=if(n==0,1,polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2,n)); \\ Joerg Arndt, May 14 2013

a(n) = C(n-1)*2^n, n>=1, a(0):=1, with C(n):=A000108(n) (Catalan).
G.f.: 1 + (2*x)*c(2*x) with c(x):=(1-sqrt(1-4*x))/(2*x), the o.g.f. of Catalan numbers A000108.
a(n) = A025225(n), n>0. - R. J. Mathar, Aug 11 2008
G.f.: (3 - sqrt(1-8*x))/2 = 2 - U(0) where U(k)=1 - 2*x/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 29 2012
G.f.: 2 - 1/Q(0), where Q(k)= 1 + (8*k+2)*x/(k+1 - x*(2*k+2)*(8*k+6)/(2*x*(8*k+6) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013

A181282 a(n) is the number of associate Rota-Baxter words in one idempotent generator x and one idempotent operator P of degree n. Such words are Rota-Baxter words that begin and/or end with x, and P is applied n times in the word.

Original entry on oeis.org

1, 3, 12, 60, 336, 2016, 12672, 82368, 549120, 3734016, 25798656, 180590592, 1278025728, 9128755200, 65727037440, 476521021440, 3475800391680, 25489202872320, 187815179059200, 1389832325038080, 10324468700282880

Offset: 0

William Sit (wyscc(AT)sci.ccny.cuny.edu), Oct 11 2010

nonn

			For n = 2, the a(2) = 12 associate Rota-Baxter words are: xP(xP(x)), xP(xP(x))x, P(xP(x))x, xP(P(x)x), xP(P(x)x)x, P(P(x)x)x, xP(xP(x)x), xP(xP(x)x)x, P(xP(x)x)x, xP(x)xP(x), xP(x)xP(x)x, P(x)xP(x)x.

Indranil Ghosh, Table of n, a(n) for n = 0..1000
L. Guo and W. Sit, Enumeration and generating functions of Rota-Baxter words, Math. Comp. Sci. (2010). In G. Regensburger, M. Rosenkranz, and W. Sit, eds., Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Sp. Issue, Math. C. Sc., 4 (2,3) (2010).

Cf. A000108, A003645, A025225, A000257.

[1] cat [3*2^(n-1)*Catalan(n): n in [1..40]]; // G. C. Greubel, Jan 04 2023

CoefficientList[Series[(3-4x-3Sqrt[1-8x])/(8x), {x,0,40}], x]
a[0] = 1; a[n_]:= 3*2^(n-1) CatalanNumber[n]; Table[a[n], {n,0,20}] (* Indranil Ghosh, Mar 05 2017 *)

a(n) = if(n==0, 1, 3*2^(n-1)*(binomial(2*n,n)/(n+1))); \\ Indranil Ghosh, Mar 05 2017

import math
f = math.factorial
def C(n,r): return f(n)/f(r)/f(n-r)
def A181282(n): return 1 if n==0 else 3*2**(n-1)*(C(2*n,n)/(n+1)) # Indranil Ghosh, Mar 05 2017

[3*2^(n-1)*catalan_number(n) -int(n==0)/2 for n in range(41)] # G. C. Greubel, Jan 04 2023

a(n) = 3*2^(n-1)*A000108(n).
G.f.: (3 - 4*t - 3*sqrt(1-8*t))/(8*t).
(n+1)*a(n) = 4*(2*n-1)*a(n-1). - R. J. Mathar, Jul 24 2012
a(n) = (n+2) * A000257(n). - F. Chapoton, Feb 26 2024

A371655 G.f. satisfies A(x) = 1 + x * A(x) * (1 + A(x))^2.

Original entry on oeis.org

1, 4, 32, 336, 4032, 52352, 716032, 10161408, 148229120, 2208921600, 33482670080, 514630230016, 8001860567040, 125640146354176, 1989285578473472, 31725578742464512, 509178657425326080, 8217766225008656384, 133287551280741351424, 2171450128344786403328

Offset: 0

Seiichi Manyama, Apr 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..806

Cf. A106228, A107708, A346626.
Cf. A025225, A371658, A371661.
Cf. A100327.

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 4^(n-k)*binomial(n, k)*binomial(2*n-k, n-1-2*k))/n);

a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 4^(n-k) * binomial(n,k) * binomial(2*n-k,n-1-2*k) for n > 0.

a(n) = 2^n * A100327(n). - Seiichi Manyama, Dec 26 2024

A103973 Expansion of (sqrt(1-8x^2)+8x^2+2x-1)/(2xsqrt(1-8x^2)).

Original entry on oeis.org

1, 2, 4, 4, 24, 16, 160, 80, 1120, 448, 8064, 2688, 59136, 16896, 439296, 109824, 3294720, 732160, 24893440, 4978688, 189190144, 34398208, 1444724736, 240787456, 11076222976, 1704034304, 85201715200, 12171673600, 657270374400

Offset: 0

Paul Barry, Feb 23 2005

Cf. A025225, A059304.

G.f.: 1/sqrt(1-8*x^2)+(1-sqrt(1-8*x^2))/(2*x).
a(n) = sum{k=0..floor(n/2), 2^(n-k) * A000108(k) * C(k+1, n-k)}.
Conjecture D-finite with recurrence: 11*n*(n+1)*a(n)+4*n*(4*n+1)*a(n-1) +8*(27-11*n^2)*a(n-2) -32*(4*n+9)*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 09 2012
a(n) ~ 2^((3*n + 1)/2) / sqrt(Pi*n) if n is even and a(n) ~ 2^((3*n + 2)/2) / (sqrt(Pi)*n^(3/2)) if n is odd. - Vaclav Kotesovec, Nov 19 2021

cp's OEIS Frontend

A141318 Euler transform of A025225.

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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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A052713 Expansion of e.g.f. (1-sqrt(1-8*x))/2.

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A371658 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x))^2.

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A100225 G.f. A(x) satisfies: 3^n - 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (3+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

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A100238 G.f. A(x) satisfies: 2^n + 1 = Sum_{k=0..n} [x^k] A(x)^n for n>=1.

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A115125 A sequence related to Catalan numbers A000108.

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A103973 Expansion of (sqrt(1-8x^2)+8x^2+2x-1)/(2xsqrt(1-8x^2)).