A025227 -id:A025227 - cp's OEIS frontend

A333473 a(n) = [x^n] ( S(x/(1 + x)) )^n, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2x) is the o.g.f. of the large Schröder numbers A006318.

1, 2, 12, 92, 752, 6352, 54768, 478928, 4231424, 37680320, 337622912, 3040354176, 27492359936, 249463806464, 2270319909632, 20714443816192, 189418898063360, 1735482632719360, 15928224355854336, 146414296847992832, 1347721096376573952, 12421053168197722112

Offset: 0

Peter Bala, Mar 23 2020

Let F(x) = 1 + f(1)*x + f(2)*x^2 + ... be a power series with integer coefficients. The associated sequence s(n) := [x^n] F(x)^n is known to satisfy the Gauss congruences: s(n*p^k) == s(n*p^(k-1)) ( mod p^(k) ) for any prime p and positive integers n and k. For certain power series F(x) we may get stronger congruences. Examples include F(x) = (1 + x)^2, F(x) = 1/(1 - x) and F(x) = c(x), where c(x) is the o.g.f. of the Catalan numbers A000108. The associated sequences (with some differences of offset) are A000984, A001700 and A025174, respectively.
Here we take F(x) = S(x/(1 + x)) = 1 + 2*x + 4*x^2 + 12*x^3 + 40*x^4 + 154*x^5 + 544*x^6 + ...(see A025227), where S(x) is the o.g.f. of the large Schröder numbers A006318. We conjecture that the associated sequence a(n) = [x^n] ( S(x/(1 + x)) )^n satisfies the congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for prime p >= 5 and positive integers n and k. Cf. A333472.
More generally, we conjecture that for a positive integer r and integer s, the sequence a(r,s;n) := [x^(r*n)] ( S(x/(1 + x)) )^(s*n) also satisfies the above congruences.
Note the sequence b(n) := [x^n] ( S(x) )^n = A103885(n) appears to satisfy the stronger congruences b(n*p^k) == b(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. The sequence d(n) := [x^n] ( (1 + x)*S(x/(1 + x)) )^n = A333090(n) also appears to satisfy the same congruences.

			Examples of congruences:
a(11) - a(1) = 3040354176 - 2 = 2*(11^2)*13*966419 == 0 ( mod 11^2 ).
a(3*7) - a(3) = 12421053168197722112 - 92 = (2^2)*(3^7)*5*(7^2)* 5795401942927 == 0 ( mod 7^2 ).
a(5^2) - a(5) = 90551762251592215396352 - 6352 = (2^4)*(5^4)*293* 30905038311123623 == 0 ( mod 5^4 ).

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

Main diagonal of A378317.

Cf. A006318, A025227, A103885, A333090, A333472.

Sch := x -> (1/2)*(1-x-sqrt(1-6*x+x^2))/x:
G := x → Sch(x/(1+x));
H := (x, n) -> series(G(x)^n, x, 51):
seq(coeff(H(x, n), x, n), n = 0..25)

Table[SeriesCoefficient[((1 - Sqrt[1- 4*x - 4*x^2])/(2*x))^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = sum(k=0, n, binomial(n, k)*binomial(n+2*k-1, 2*k)); \\ Seiichi Manyama, Nov 24 2024

a(n) = [x^n] ( (1 - sqrt(1- 4*x - 4*x^2))/(2*x) )^n.
a(n) ~ sqrt(((sqrt(2) + 1)^(2/3) + (sqrt(2) - 1)^(2/3) - 1)/3) * ((3*(71 + 8*sqrt(2))^(1/3) + 3*(71 - 8*sqrt(2))^(1/3) + 13))^n / (sqrt(Pi*n) * 2^(2*n+1)). - Vaclav Kotesovec, Mar 28 2020
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+2*k-1,2*k). - Seiichi Manyama, Nov 24 2024

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

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

Original entry on oeis.org

1, 1, 18, 333, 6318, 122634, 2429028, 48974949, 1002875094, 20814628158, 437088964860, 9272342710962, 198456435657036, 4280758166952756, 92972201833888200, 2031520673763657621, 44630859892110807654

Offset: 0

Wolfdieter Lang, Mar 04 2002

Fung Lam, Table of n, a(n) for n = 0..725

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

a[n_] := (288 (2 - n) a[n - 2] + 18 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}](* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-36*x*(1-8*x)])/(18*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n) = (9^n) * p(n, -8/9) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 9*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-36*x*(1-8*x)))/(18*x).
Recurrence: (n+1)*a(n) = 288*(2-n)*a(n-2) + 18*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(2) * 24^n / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

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

A182399 G.f. A(x) satisfies: A(A(x)) - A(A(x))^2 = x + x^2.

Original entry on oeis.org

1, 1, 1, 3, 7, 21, 61, 187, 583, 1837, 5885, 19027, 62167, 204917, 680621, 2275211, 7648519, 25852573, 87812093, 299349795, 1023570647, 3515918501, 12140103149, 41894710427, 143835281351, 501071173901, 1808088546557, 6212411239539, 17720665594455

Offset: 1

Paul D. Hanna, Apr 27 2012

sign

a(33) is the first negative term.

If B(x) = x + 2*x^2 + 8*x^3 + 36*x^4 + 160*x^5 + 736*x^6 + 3648*x^7 + ..., then g.f. A(x) = x + B(x * A(x)). - Michael Somos, Jun 27 2017

			G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 7*x^5 + 21*x^6 + 61*x^7 + 187*x^8 +...
Related expansions:
A(A(x)) = x + 2*x^2 + 4*x^3 + 12*x^4 + 40*x^5 + 144*x^6 + 544*x^7 + 2128*x^8 +...
A(A(x))^2 = x^2 + 4*x^3 + 12*x^4 + 40*x^5 + 144*x^6 + 544*x^7 + 2128*x^8 +...
where A(A(x)) - A(A(x))^2 = x + x^2.
Let C(x) satisfy C(x-x^2) = x, where C(x) begins:
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 +...+ A000108(n-1)*x^n +...
then
A(-C(-x)) = x + x^3 + 4*x^5 + 21*x^7 + 122*x^9 + 758*x^11 + 4958*x^13 +...+ (-1)^(n-1)*A179270(2*n-1)*x^(2*n-1) +...

Paul D. Hanna, Table of n, a(n) for n = 1..256

Cf. A025227, A179270, A000108.

T(n, m):= if n=m then 1 else ((sum((binomial(k+m,n-k-m)*binomial(2*k+m-1,k+m-1))/(k+m),k,0,n-m))*m -sum(T(n, i) *T(i, m), i, m+1, n-1))/2;
makelist(T(n, 1), n, 1, 10); /* Vladimir Kruchinin, Apr 28 2012 */

{a(n)=local(A=x+x^2,G);for(i=1,n,G=subst(A,x,A+x*O(x^n));A=A+(x+x^2-G+G^2)/2);polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

/* Faster vectorized version: */
{MM=100;A=[1];B=x;C=(1-sqrt(1-4*(x+x^2+x*O(x^MM))))/2; for(n=1,oo,A=concat(A,0);B=x*Ser(A); A[n]=Vec((B+subst(C+x*O(x^n),x,serreverse(B)))/2)[n]; print1(A[n],", "))}

/* PARI/GP Version of Vladimir Kruchinin's formula: */
{T(n, m)=if(n==m,1, if(n>m, (sum(k=0,n-m,(binomial(k+m,n-k-m)*binomial(2*k+m-1,k+m-1))/(k+m))*m - sum(i=m+1,n-1,T(n, i) *T(i, m)))/2 ))}
{a(n)=T(n,1)}

G.f. satisfies: A(-A(-x)) = x.
G.f. satisfies: A(A(x)) = (1 - sqrt(1-4*(x+x^2)))/2 is the g.f. of A025227; thus, A(A(x)) = C(x+x^2) where C(x-x^2) = x.
G.f. satisfies: A(-C(-x)) = -I*G(I*x) where C(x-x^2) = x and G(x) is the g.f. of A179270 such that the inverse of function G(x) + I*G(x)^2 equals the complex conjugate: G(x) - I*G(x)^2.
a(n) = T(n,1), with T(n, m) = (sum((binomial(k+m,n-k-m)*binomial(2*k+m-1,k+m-1))/(k+m),k,0,n-m)*m -sum(T(n, i) *T(i, m), i, m+1, n-1))/2, n>m, T(n,n) = 1. - Vladimir Kruchinin, Apr 28 2012

A260774 Certain directed lattice paths.

Original entry on oeis.org

1, 6, 33, 189, 1107, 6588, 39663, 240894, 1473147, 9058554, 55954395, 346934745, 2157989445, 13459891500, 84152389833, 527224251861, 3309194474451, 20804569738218, 130987600581699, 825796890644895, 5212349717906889, 32935490120006604, 208316726580941037

Offset: 0

N. J. A. Sloane, Jul 30 2015

nonn

See Dziemianczuk (2014) for precise definition.

Alois P. Heinz, Table of n, a(n) for n = 0..1235 (first 101 terms from Lars Blomberg)
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

Cf. A122868, A064641, A156016.

Cf. A006139, A179191, A052709, A025227.

b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
      `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
      `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 21 2021

b[x_, y_] := b[x, y] = If[{x, y} == {0, 0}, 1,
     If[x > 0, Sum[b[x - 1, y + j], {j, -1, 1}], 0] +
     If[y > 0, b[x, y - 1], 0] + If[y < 0, b[x, y + 1], 0]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

See Dziemianczuk (2014) Equation (33a) with m=1.
From Vaclav Kotesovec, Jul 15 2022: (Start)
Recurrence: (n+1)*(4*n - 3)*a(n) = 6*(4*n^2 - n - 1)*a(n-1) + 3*(n-1)*(4*n + 1)*a(n-2).
a(n) ~ (3 + 2*sqrt(3))^(n+1) / sqrt(6*Pi*n). (End)

More terms from Lars Blomberg, Aug 01 2015

A366554 G.f. A(x) satisfies A(x) = 1 + x + x^4*A(x)^2.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 0, 2, 6, 6, 2, 5, 20, 30, 20, 19, 70, 140, 140, 112, 266, 630, 840, 762, 1176, 2814, 4620, 5049, 6204, 12936, 24156, 31460, 36894, 63492, 123552, 185471, 228800, 338910, 634920, 1050686, 1411410, 1944800, 3354780, 5820256, 8513804, 11644490

Offset: 0

Seiichi Manyama, Oct 13 2023

nonn

Cf. A007477, A025227, A248100.
Cf. A366556, A366558.
Cf. A366589.

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

G.f.: A(x) = 2*(1+x) / (1+sqrt(1-4*x^4*(1+x))).
a(n) = Sum_{k=0..floor(n/4)} binomial(k+1,n-4*k) * binomial(2*k,k)/(k+1).
a(n) = A366589(n) + A366589(n-1).

cp's OEIS Frontend

A333473 a(n) = [x^n] ( S(x/(1 + x)) )^n, where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the large Schröder numbers A006318.

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

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

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

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

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

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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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PARI

PARI

PARI

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A182399 G.f. A(x) satisfies: A(A(x)) - A(A(x))^2 = x + x^2.

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A333473 a(n) = [x^n] ( S(x/(1 + x)) )^n, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2x) is the o.g.f. of the large Schröder numbers A006318.

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.