A068763 -id:A068763 - cp's OEIS frontend

A025227 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 3.

0, 1, 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

Clark Kimberling

Series reversion of g.f. A(x) is -A(-x). - Michael Somos, Jul 27 2003
a(n) is the number of royal paths (A006318) from (0,0) to (n-1,n-1) such that every northeast (diagonal) step is either immediately followed by a north step or ends the path. For example a(3)=4 counts EDN, EENN, END, ENEN (E=east, D=diagonal, N=north). - David Callan, Jul 03 2006
From David Callan, Sep 25 2006: (Start)
a(n) is the number of ordered trees with n leaves in which (i) every node (= non-root non-leaf vertex) has at least 2 children and (ii) each leaf is either the leftmost or rightmost child of its parent. For example, a(3)=4 counts
              |
     /\      / \
      /\       /\
and their mirror images. (End)
From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010: (Start)
a(n+1), n >= 0, is also the number of Rota-Baxter words in one idempotent generator x and one operator of arity n.
Alternatively, a(n+1) is the number of ways of adding pairs of parentheses to a string of n x's (the number m of parentheses pairs necessarily satisfies m <= n <= 2m+1 for a nonzero count), such that no two pairs of parentheses are immediately nested and no two x's remain adjacent. (End)
a(n) is the number of colored binary trees on n-1 vertices where leaves have 2 possible colors and internal nodes have 1 color. - Alexander Burstein, Mar 07 2020

			For n=2, a(3) = 4 has the following words: x(x), (x)x, (x(x)), ((x)x) corresponding to A(1,2)=2, and A(2,2)=2. - William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010

L. Guo and W. Sit, Enumeration of Rota-Baxter Words (extended abstract), ISSAC 2006 Proceedings, 123-131. [From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010]
L. Guo and W. Sit, Enumeration of Rota-Baxter Words, to appear in Mathematics in Computer Science, Special Issue on AADIOS special session, ACA, 2009. [From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010]

Michael De Vlieger, Table of n, a(n) for n = 0..1470
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 22.
Maciej Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
Li Guo and William Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 655
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 657
Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

Cf. A052709, A052709, A100238.
Cf. A182399, A219534, A000108, A006318.
Cf. A068763.

Table[CatalanNumber[n-1] Hypergeometric2F1[(1-n)/2, -n/2, 3/2-n, -1] + KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 17 2016 *)

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

a(n) = A052709(n) + A052709(n-1).
A100238(n) = -(-1)^n*a(n), for n>1.
a(n) = Sum_{k=0..floor(n/2)} C(n-k-1)*binomial(n-k, k), where C(q)=binomial(2q, q)/(q+1) are the Catalan numbers (A000108). - Emeric Deutsch, Nov 14 2001 [{a(n+1)}A068763.%20-%20_Wolfdieter%20Lang">{n>=0} = row sum of A068763. - _Wolfdieter Lang, Jan 21 2023]
D-finite with recurrence n*a(n) = (4n-6)*a(n-1)+(4n-12)*a(n-2), n>2. a(1)=1, a(2)=2.
G.f. satisfies A(x)-A(x)^2 = x+x^2. - Ralf Stephan, Jun 30 2003
a(n) = Sum_{k=0..n-1} C(k)*C(k+1, n-k-1). - Paul Barry, Feb 23 2005
G.f. A(x) satisfies A(x)=x+C(2x*A(x)) where C(x) is g.f. of Catalan numbers A000108 offset 1. - Michael Somos, Sep 08 2005
G.f.: (1-sqrt(1-4x-4x^2))/2 = 2(x+x^2)/(1+sqrt(1-4x-4x^2)). - Michael Somos, Jun 08 2000
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example Phi([1]) is the Catalan numbers A000108. The present sequence is (essentially) Phi([1,2]). - Gary W. Adamson, Oct 27 2008
From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010: (Start)
a(n+1), n >= 0, is column sum for the n-th column of the table R(m,n)=binomial(m+1, n-m)c(m) where c(m) is the m-th Catalan number A000108.
The table entry is nonzero if and only if m <= n <= 2m+1.
R(m,n) gives the number of Rota-Baxter words in one idempotent generator x and one operator of degree m and arity n, or the number of ways of adding m pairs of parentheses to a string of n x's (n necessarily lies between m and 2m+1 inclusive for a nonzero count), such that no two pairs of parentheses are immediately nested and no two x's remain adjacent. (End)
G.f.: A(x) = B(B(x)) where B(x) is the g.f. of A182399. -Paul D. Hanna, Apr 27 2012
G.f.: 1 - x + x*G(0), where G(k) = 1 + 1/(1 - (1+x)/(1 + x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 01 2013
a(n) ~ (1 + sqrt(2))^(n - 1/2) * 2^(n - 5/4) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 18 2013, simplified Jan 21 2023
O.g.f.: A(x) = x*S(x/(1 + x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318. - Peter Bala, Mar 05 2020
G.f.: A(x) satisfies ((A(x) - A(-x))/(2*x))^2 = S(4*x^2), where S(x) is the g.f. for the large Schröder numbers A006318. - Alexander Burstein, May 20 2021
A(x) = (x + x^2)*c(x+x^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Note that (x - x^2)*c(x-x^2) = x. - Peter Bala, Aug 29 2024

A068764 Generalized Catalan numbers 2xA(x)^2 -A(x) +1 -x =0.

Original entry on oeis.org

1, 1, 4, 18, 88, 456, 2464, 13736, 78432, 456416, 2697088, 16141120, 97632000, 595912960, 3665728512, 22703097472, 141448381952, 885934151168, 5575020435456, 35230798994432, 223485795258368, 1422572226146304, 9083682419818496, 58169612565614592, 373486362257899520, 2403850703479816192

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n) = K(2,2; n)/2 with K(a,b; n) defined in a comment to A068763.

Hankel transform is A166232(n+1). - Paul Barry, Oct 09 2009

			G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 456*x^5 + 2464*x^6 + 13736*x^7 + ...

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

Cf. A025227, A025228, A025229, A025230.

Cf. A071356, A001003, A025235.

Table[SeriesCoefficient[(1-Sqrt[1-8*x*(1-x)])/(4*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
Round@Table[4^(n-1) Hypergeometric2F1[(1-n)/2, 1-n/2, 2, 1/2] + KroneckerDelta[n]/Sqrt[2], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4^(n - 1) Hypergeometric2F1[ (1 - n)/2, (2 - n)/2, 2, 1/2]]; (* Michael Somos, Nov 08 2015 *)

a(n):=sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n); /* Vladimir Kruchinin, Aug 11 2010 */

{a(n) = my(A); if( n<1, n==0, n--;  A = x * O(x^n); n! * simplify( polcoeff( exp(4*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* Michael Somos, Mar 31 2007 */

x='x+O('x^66); Vec((1-sqrt(1-8*x*(1-x)))/(4*x)) \\ Joerg Arndt, May 06 2013

G.f.: (1-sqrt(1-8*x*(1-x)))/(4*x).
a(n+1) = 2*sum(a(k)*a(n-k), k=0..n), n>=1, a(0) = 1 = a(1).
a(n) = (2^n)*p(n, -1/2) with the row polynomials p(n, x) defined from array A068763.
E.g.f. (offset -1) is exp(4*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic, Mar 31 2004
The o.g.f. satisfies A(x) = 1 + x*(2*A(x)^2 - 1), A(0) = 1. - Wolfdieter Lang, Nov 13 2007
a(n) = subs(t=1,(d^(n-1)/dt^(n-1))(-1+2*t^2)^n)/n!, n >= 2, due to the Lagrange series for the given implicit o.g.f. equation. This formula holds also for n=1 if no differentiation is used. - Wolfdieter Lang, Nov 13 2007, Feb 22 2008
1/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-..... (continued fraction). - Paul Barry, Jan 29 2009
a(n) = A166229(n)/(2-0^n). - Paul Barry, Oct 09 2009
a(n) = sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n), n>0. - Vladimir Kruchinin, Aug 11 2010
D-finite with recurrence: (n+1)*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
a(n) = 4^(n-1)*hypergeom([(1-n)/2,1-n/2], [2], 1/2) + 0^n/sqrt(2). - Vladimir Reshetnikov, Nov 07 2015
0 = a(n)*(+64*a(n+1) - 160*a(n+2) + 32*a(n+3)) + a(n+1)*(+32*a(n+1) + 48*a(n+2) - 20*a(n+3)) + a(n+2)*(+4*a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Nov 08 2015
a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(2*k+1,n) / (2*k+1). - Seiichi Manyama, Jul 24 2023

A068772 Generalized Catalan numbers 10xA(x)^2 -A(x) +1 -9*x =0.

Original entry on oeis.org

1, 1, 20, 410, 8600, 184200, 4020000, 89205000, 2008700000, 45816140000, 1056825200000, 24618524200000, 578457724000000, 13695679012000000, 326448619920000000, 7827776361090000000, 188701194087000000000

Offset: 0

Wolfdieter Lang, Mar 04 2002

This is the tenth member in the a-family of sequences K(a,a; n), a=1,2,3,...,n>=0, defined in a comment to the array A068763.

Fung Lam, Table of n, a(n) for n = 0..700

Cf. A000108, A068764-A068771, A025227-A025230.

a[0] = 1; a[1] = 1; a[n_] := (360 (2 - n) a[n - 2] + 20 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-40*x*(1-9*x)])/(20*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2014 *)

a(n) = (10^n) * p(n, -9/10) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 10*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-40*x*(1-9*x)))/(20*x).
Recurrence: (n+1)*a(n) = 360*(2-n)*a(n-2) + 20*(2*n-1)*a(n-1). - Fung Lam, Mar 05 2014
a(n) ~ sqrt(5+5*sqrt(10)) * (20+2*sqrt(10))^n / (10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2014

A068766 Generalized Catalan numbers 4xA(x)^2 -A(x)+1-3*x=0.

Original entry on oeis.org

1, 1, 8, 68, 608, 5664, 54528, 538944, 5441024, 55889408, 582348800, 6140864512, 65414742016, 702897995776, 7609805045760, 82929151328256, 908978855215104, 10014523823357952, 110840574196580352, 1231847926116384768

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n)=K(4,4; n)/4 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..925

Cf. A000108, A068764-5, A068767-72, A025227-30.

a := n -> `if`(n=0,1,simplify(2^n*GegenbauerC(n-1, -n, -2))/(2*n)):
seq(a(n), n=0..19); # Peter Luschny, May 09 2016

CoefficientList[Series[(1-Sqrt[1-16*x*(1-3*x)])/(8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n)=(4^n)*p(n, -3/4) with the row polynomials p(n, x) defined from array A068763.
a(n+1)= 4*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-16*x*(1-3*x)))/(8*x).
Recurrence: (n+1)*a(n) = 48*(2-n)*a(n-2) + 8*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(6) * 12^n / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014
a(n) = 2^n*GegenbauerC(n-1, -n, -2)/(2*n) for n>=1. - Peter Luschny, May 09 2016

A068767 Generalized Catalan numbers 5xA(x)^2 -A(x) +1 -4*x=0.

Original entry on oeis.org

1, 1, 10, 105, 1150, 13050, 152500, 1825625, 22293750, 276758750, 3483287500, 44352006250, 570333187500, 7396680812500, 96638930625000, 1270796364765625, 16806545339843750, 223400240246093750

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n) = K(5,5; n)/5 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..860

Cf. A000108, A068764-A068766, A068768-A068772, A025227-A025230.

CoefficientList[Series[(1-Sqrt[1-20*x*(1-4*x)])/(10*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n) = (5^n) * p(n, -4/5) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 5*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-20*x*(1-4*x)))/(10*x).
(n+1)*a(n) = 80*(2-n)*a(n-2) + 10*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(10+10*sqrt(5)) * (10+2*sqrt(5))^n / (10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014
Equivalently, a(n) ~ 2^(2*n) * 5^((n-1)/2) * phi^(n + 1/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

A068768 Generalized Catalan numbers 6xA(x)^2 -A(x) +1 -5*x =0.

Original entry on oeis.org

1, 1, 12, 150, 1944, 25992, 356832, 5008824, 71629920, 1040509152, 15315578496, 227981324736, 3426473187072, 51929043390720, 792725911280640, 12178706839758720, 188158789025809920, 2921622674591946240

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n) = K(6,6; n)/6 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..810

Cf. A000108, A068764-A068767, A068769-A068772, A025227-A025230.

CoefficientList[Series[(1-Sqrt[1-24*x*(1-5*x)])/(12*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n) = (6^n) * p(n, -5/6) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 6*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-24*x*(1-5*x)))/(12*x).
D-finite with recurrence: (n+1)*a(n) = 120*(2-n)*a(n-2) + 12*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(3+3*sqrt(6)) * (12+2*sqrt(6))^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

A068769 Generalized Catalan numbers 7xA(x)^2 -A(x) +1 -6*x=0.

Original entry on oeis.org

1, 1, 14, 203, 3038, 46746, 736764, 11853051, 194053622, 3224557406, 54265836548, 923218762270, 15854602773100, 274500192707860, 4786546243533432, 83989334625037947, 1481965556616225702

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n) = K(7,7; n)/7 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..780

Cf. A000108, A068764-A068768, A068770-A068772, A025227-A025230.

a[0] = 1; a[1] = 1; a[2] = 14; a[n_] := (168 (2 - n) a[n - 2] + 14 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-28*x*(1-6*x)])/(14*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n) = (7^n) * p(n, -6/7) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 7*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-28*x*(1-6*x)))/(14*x).
Recurrence: (n+1)*a(n) = 168*(2-n)*a(n-2) + 14*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(14+14*sqrt(7)) * (14+2*sqrt(7))^n / (14*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

A068770 Generalized Catalan numbers 8xA(x)^2 -A(x) +1 -7*x=0.

Original entry on oeis.org

1, 1, 16, 264, 4480, 77952, 1386496, 25135616, 463233024, 8658673664, 163829383168, 3132565553152, 60446638866432, 1175715287400448, 23028562592268288, 453848132868898816, 8993594212565909504

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n) = K(8,8; n)/8 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..750

Cf. A000108, A068764-A068769, A068771-A068772, A025227-A025230.

CoefficientList[Series[(1-Sqrt[1-32*x*(1-7*x)])/(16*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n) = if(n, (4^(n-1)*14^(1/2*n+1/2)*pollegendre(n+1,2/7*14^(1/2)) - pollegendre(n,2/7*14^(1/2))*4^n*14^(n/2))\/n, 1) \\ Charles R Greathouse IV, Mar 19 2017

a(n) = (8^n) * p(n, -7/8) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 8*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-32*x*(1-7*x)))/(16*x).
a(n) = (4^(n-1)*14^(1/2*n+1/2)*LegendreP(n+1,2/7*14^(1/2)) - LegendreP(n,2/7*14^(1/2))*4^n*14^(1/2*n))/n for n > 0. - Mark van Hoeij, Apr 23 2010
Recurrence: (n+1)*a(n) = 224*(2-n)*a(n-2) + 16*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(1+2*sqrt(2)) * (16+4*sqrt(2))^n / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

cp's OEIS Frontend

A025227 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A068764 Generalized Catalan numbers 2*x*A(x)^2 -A(x) +1 -x =0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

PARI

Formula

A068772 Generalized Catalan numbers 10*x*A(x)^2 -A(x) +1 -9*x =0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A068766 Generalized Catalan numbers 4*x*A(x)^2 -A(x)+1-3*x=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A068767 Generalized Catalan numbers 5*x*A(x)^2 -A(x) +1 -4*x=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A068768 Generalized Catalan numbers 6*x*A(x)^2 -A(x) +1 -5*x =0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A068769 Generalized Catalan numbers 7*x*A(x)^2 -A(x) +1 -6*x=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A068770 Generalized Catalan numbers 8*x*A(x)^2 -A(x) +1 -7*x=0.

A025227 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 3.

A068764 Generalized Catalan numbers 2xA(x)^2 -A(x) +1 -x =0.

A068772 Generalized Catalan numbers 10xA(x)^2 -A(x) +1 -9*x =0.

A068766 Generalized Catalan numbers 4xA(x)^2 -A(x)+1-3*x=0.

A068767 Generalized Catalan numbers 5xA(x)^2 -A(x) +1 -4*x=0.

A068768 Generalized Catalan numbers 6xA(x)^2 -A(x) +1 -5*x =0.

A068769 Generalized Catalan numbers 7xA(x)^2 -A(x) +1 -6*x=0.

A068770 Generalized Catalan numbers 8xA(x)^2 -A(x) +1 -7*x=0.

A068771 Generalized Catalan numbers 9xA(x)^2 -A(x) +1 -8*x=0.

A068765 Generalized Catalan numbers 3xA(x)^2 -A(x) +1 -2*x = 0.