A376636 -id:A376636 - cp's OEIS frontend

A004987 a(n) = (3^n/n!)Product_{k=0..n-1} (3k + 1). 3-central binomial coefficients.

1, 3, 18, 126, 945, 7371, 58968, 480168, 3961386, 33011550, 277297020, 2344420260, 19927572210, 170150808870, 1458435504600, 12542545339560, 108179453553705, 935434098376155, 8107095519260010, 70403724246205350, 612512400941986545, 5337608065351597035, 46582761297613937760

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Diagonal of rational function R(x,y) = (1 - 9*x*y) / (1 - 2*x - 3*y + 3*y^2 + 9*x^2*y). - Gheorghe Coserea, Jul 01 2016

This is the k = 3 variant of the k-central binomial coefficients c(n,k) with g.f. (1 - k^2*x)^(-1/k), which yield the usual central binomial coefficients A001405 for k = 2. - M. F. Hasler, Nov 12 2024

			G.f.: 1 + 3*x + 18*x^2 + 126*x^3 + 945*x^4 + 7371*x^5 + 58968*x^6 + 480168*x^7 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..200
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
A. Straub, V. H. Moll, and T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (2009) 31-41, eq (1.10).

Cf. A004117, A007559, A047657, A004988, A054861, A034689, A053101, A072888, A216316.
Related to diagonal of rational functions: A268545-A268555.
Cf. A008931, A078532, A245114, A247029, A376282, A376636.

List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k+1)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

[1] cat [3^n*&*[3*k+1: k in [0..n-1]]/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019

a:= n-> (3^n/n!)*mul(3*k+1, k=0..n-1); seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019

Table[(-9)^n Binomial[-1/3, n], {n, 0, 25}] (* Jean-François Alcover, Sep 28 2016, after Peter Luschny *)

a(n) = prod(k=0, n-1, 3*k + 1)*3^n/n! \\ Michel Marcus, Jun 30 2013

my(x='x, y='y);
R = (1 - 9*x*y) / (1 - 2*x - 3*y + 3*y^2 + 9*x^2*y);
diag(n, expr, var) = {
  my(a = vector(n));
  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
  for (k = 1, n, a[k] = expr;
       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
  return(a);
};
diag(20, R, [x,y])  \\ Gheorghe Coserea, Jul 01 2016

Vec((1-9*x+O(x^25))^(-1/3)) \\ yields the same as:
apply( {A004987(n)=prod(k=0, n-1, 9*k+3)\n!}, [0..24]) \\ M. F. Hasler, Nov 12 2024

[9^n*rising_factorial(1/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019

G.f.: (1 - 9*x)^(-1/3).
a(n) = (3^n/n!)*A007559(n), n >= 1, a(0) := 1.
a(n) ~ Gamma(1/3)^-1*n^(-2/3)*3^(2*n)*{1 - 1/9*n^-1 + ...}.
Representation as n-th moment of a positive function on (0, 9): a(n) = Integral_{x=0..9} ( x^n*(1/(Pi*sqrt(3)*6*(x/9)^(2/3)*(1-x/9)^(1/3))) ), n >= 0. This function is the solution of the Hausdorff moment problem on (0, 9) with moments equal to a(n). As a consequence this representation is unique. - Karol A. Penson, Jan 30 2003
D-finite with recurrence: n*a(n) + 3*(2-3*n)*a(n-1)=0. - R. J. Mathar, Jun 07 2013
0 = a(n) * (81*a(n+1) - 15*a(n+2)) + a(n+1) * (-3*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Jan 27 2014
G.f. A(x)=:y satisfies 0 = y'' * y - 4 * y' * y'. - Michael Somos, Jan 27 2014
a(n) = (-9)^n*binomial(-1/3, n). - Peter Luschny, Mar 23 2014
E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([1/3], [1], 9*x). - Karol A. Penson, Dec 19 2015
Sum_{n>=0} 1/a(n) = (sqrt(3)*Pi + 3*(12 + log(3)))/32 = 1.3980385924595932... - Ilya Gutkovskiy, Jul 01 2016
Binomial transform of A216316. - Peter Bala, Jul 02 2023
From Peter Bala, Mar 31 2024: (Start)
a(n) = (9^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-1/3, k)* binomial(-1/3, 2*n - k).
(9^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
Sum_{k = 0..n} a(k)*a(n-k) = A004988(n).
Sum_{k = 0..2*n} a(k)*a(2*n-k) = 18^n/(2*n)! * Product_{k = 1..n} (6*k - 1)*(3*k - 2). (End)
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^3). - Seiichi Manyama, Jun 20 2025

More terms from Ralf Stephan, Mar 13 2004

More terms from Benoit Cloitre, Jun 05 2004

A078532 Coefficients of power series that satisfies A(x)^3 - 9xA(x)^4 = 1, A(0)=1.

Original entry on oeis.org

1, 3, 27, 315, 4158, 59049, 880308, 13586859, 215233605, 3479417370, 57168561996, 951892141473, 16026585711660, 272383068872700, 4666865660812044, 80521573261807755, 1397858693681272230, 24398716826612190447, 427921056863230599900, 7537621933880388620010

Offset: 0

Paul D. Hanna, Nov 28 2002

If A(x) = Sum_{k>=1} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2) (conjecture).
If A(x) = Sum_{k>=1} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k)=n^(2k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - Emeric Deutsch, Dec 10 2002
A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n -(n^2)*x*A(x)^(n+1) = 1 reduces to A(x)=1+xA(x)^2. - Emeric Deutsch, Dec 10 2002
Radius of convergence of g.f. A(x) is r = 1/(3*4^(4/3)) where A(r) = 4^(1/3). - Paul D. Hanna, Jul 24 2012
Self-convolution cube yields A214668.

			A(x)^3 - 9x*A(x)^4 = 1 since A(x)^3 = 1 +9x +108x^2 +1458x^3 +21060x^4 +... and A(x)^4 = 1 +12x +162x^2 +2340x^3 +... also a(2)=3^3, a(5)=3^10.

G. C. Greubel, Table of n, a(n) for n = 0..750

Cf. A214668, A078531, A078533, A078534, A078535.
Cf. A004987, A008931, A245114, A247029, A376282, A376636.
Cf. A004990.

Table[3^(2n) Binomial[(4n-2)/3,n]/(n+1),{n,0,20}] (* Harvey P. Dale, Nov 03 2011 *)

for(n=0,25, print1(9^n * binomial((4*n-2)/3, n)/(n+1), ", ")) \\ G. C. Greubel, Jan 26 2017

a(n) = 3^(2n)*binomial(4n/3-2/3, n)/(n+1). - Emeric Deutsch, Dec 10 2002
Sequence with offset 1 is expansion of reversion of g.f. x*(1-9*x)^(1/3), which equals x times the g.f. of A004990.
a(n) ~ 2^(8*n/3-5/6) * 3^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 03 2014
D-finite with recurrence n*(n-1)*(n+1)*a(n) -216*(4*n-5)*(2*n-1)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Mar 24 2023
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^5). - Seiichi Manyama, Jun 20 2025

More terms from Harvey P. Dale, Nov 03 2011

A245114 G.f. A(x) satisfies A(x)^3 = 1 + 9xA(x)^5.

Original entry on oeis.org

1, 3, 36, 585, 10935, 221697, 4740120, 105225318, 2402040420, 56029889025, 1329627118248, 31998624800220, 779102941714461, 19157195459506230, 475034438632316400, 11865382635213387504, 298265217964573747095, 7539795161286074350785, 191548870595159091038640, 4888023169106780049244275

Offset: 0

Paul D. Hanna, Jul 31 2014

			G.f.: A(x) = 1 + 3*x + 36*x^2 + 585*x^3 + 10935*x^4 + 221697*x^5 +...
where A(x)^3 = 1 + 9*x*A(x)^5:
A(x)^3 = 1 + 9*x + 135*x^2 + 2430*x^3 + 48195*x^4 + 1015740*x^5 +...
A(x)^5 = 1 + 15*x + 270*x^2 + 5355*x^3 + 112860*x^4 + 2480058*x^5 +...

Robert Israel, Table of n, a(n) for n = 0..696

Cf. A004987, A008931, A078532, A247029, A376282, A376636.

Cf. A245112, A385119.

rec:= 2*a(n+3)*(n+3)*(n+2)*(n+1)*(2*n+7)=135*a(n)*(5*n+1)*(5*n+4)*(5*n+7)*(5*n+13):
f:= gfun:-rectoproc({rec,a(0)=1,a(1)=3,a(2)=36},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jan 30 2018

nmax = 19; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^3 - (1 + 9 x A[x]^5) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. HoldPattern[a[n_] -> k_] :> Set[a[n], k];
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)

/* From A(x)^3 = 1 + 9*x*A(x)^5 : */
{a(n) = local(A=1+x);for(i=1,n,A=(1 + 9*x*A^5 +x*O(x^n))^(1/3));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

{a(n) = 9^n * binomial((5*n - 2)/3, n) / (2*n+1)}
for(n=0,20,print1(a(n),", "))

a(n) = 9^n * binomial((5*n - 2)/3, n) / (2*n + 1).
2*a(n+3)*(n+3)*(n+2)*(n+1)*(2*n+7)=135*a(n)*(5*n+1)*(5*n+4)*(5*n+7)*(5*n+13). - Robert Israel, Jan 30 2018
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^7). - Seiichi Manyama, Jun 20 2025
a(n) ~ 3^n * 5^(5*n/3-1/6) / (sqrt(Pi) * 2^(2*(n+2)/3) * n^(3/2)). - Amiram Eldar, Sep 02 2025

A008931 Expansion of (2/(1+sqrt(1-36*x)))^(1/3).

Original entry on oeis.org

1, 3, 45, 936, 22572, 592515, 16434495, 473825700, 14058408519, 426438391743, 13164565835421, 412255067017248, 13064028812911440, 418149414542496168, 13498863325944967656, 439006511643775469856, 14369623854340007790108, 473027210589699351461700

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..640

Cf. A004987, A078532, A245114, A247029, A376282, A376636.

a:=[1];; for n in [2..20] do a[n]:=6*(5-21*(n-1)+18*(n-1)^2)*a[n-1]/((n-1)*(3*n-2)); od; a; # G. C. Greubel, Sep 13 2019

I:=[1]; [n le 1 select I[n] else 6*(5-21*(n-1)+18*(n-1)^2)*Self(n-1)/((n-1)*(3*n-2)): n in [1..20]]; // G. C. Greubel, Sep 13 2019

seq(9^n*binomial(2*n +1/3, n)/(6*n+1), n=0..20); # G. C. Greubel, Sep 13 2019

CoefficientList[Series[Surd[2/(1+Sqrt[1-36x]),3],{x,0,20}],x] (* Harvey P. Dale, Aug 12 2016 *)
Table[9^n Binomial[2 n + 1/3, n]/(6 n + 1), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 12 2016 *)

my(x='x+O('x^20)); Vec((2/(1+sqrt(1-36*x)))^(1/3)) \\ G. C. Greubel, Apr 11 2017

[9^n*binomial(2*n +1/3, n)/(6*n+1) for n in (0..20)] # G. C. Greubel, Sep 13 2019

From Vladimir Reshetnikov, Oct 12 2016: (Start)
a(n) = 9^n*binomial(2*n + 1/3, n)/(6*n + 1).
D-finite with recurrence: n*(3*n+1)*a(n) = 6*(18*n^2-21*n+5)*a(n-1). (End)
a(n) ~ 2^(2*n-2/3)*3^(2*n-1)/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Oct 13 2016
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^9). - Seiichi Manyama, Jun 20 2025
G.f.: 2F1(1/6, 2/3 ; 4/3 ; 36*x). - R. J. Mathar, Jul 30 2025

A376282 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^7)^(1/3).

Original entry on oeis.org

1, 3, 54, 1368, 40365, 1299078, 44223732, 1565864784, 57079952046, 2127818007315, 80742077597610, 3108398557803480, 121107814518484872, 4766365291226837508, 189209375036491438800, 7567095678024459993120, 304603864960375133224533, 12331716699093681951702810

Offset: 0

Seiichi Manyama, Oct 23 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 0..600

Cf. A004987, A008931, A078532, A245114, A376636.

A376282[n_] := 9^n*Binomial[(7*n + 1)/3, n]/(7*n + 1);
Array[A376282, 20, 0] (* Paolo Xausa, Aug 04 2025 *)

a(n) = 9^n*binomial(7*n/3+1/3, n)/(7*n+1);

a(n) = 9^n * binomial(7*n/3 + 1/3,n)/(7*n+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^11). - Seiichi Manyama, Jun 20 2025
D-finite with recurrence 8*n*(n-1)*(n-2)*(4*n-5)*(2*n-1)*(4*n+1)*a(n) -189*(7*n-11)*(7*n-17)*(7*n-2)*(7*n-20)*(7*n-5)*(7*n-8)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A247029 G.f. A(x) satisfies A(x) = A(x)^4 - 9*x.

Original entry on oeis.org

1, 3, -18, 180, -2187, 29484, -424116, 6377292, -99034650, 1576075644, -25569752274, 421325812440, -7031733125508, 118620405322020, -2019349799669160, 34647126360607440, -598525520999144643, 10401492640172342940, -181721630178565389900, 3189811189331825319492

Offset: 0

Paul D. Hanna, Sep 09 2014

sign

			G.f.: A(x) = 1 + 3*x - 18*x^2 + 180*x^3 - 2187*x^4 + 29484*x^5 - 424116*x^6 +...
where
A(x)^4 = 1 + 12*x - 18*x^2 + 180*x^3 - 2187*x^4 + 29484*x^5 - 424116*x^6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A004987, A008931, A078532, A245114, A376282, A376636.

Cf. A224884.

FullSimplify[Table[-(-1)^n * 3^(2*n-1) * 4^(n-1) * Gamma[n/3 + 1/6] * Gamma[2*n/3 - 1/6] / (Pi * Gamma[n + 1]), {n, 0, 20}]] (* Vaclav Kotesovec, Nov 18 2017 *)

{a(n)=polcoeff(x/serreverse(x*(1+9*x +x^2*O(x^n))^(1/3)), n)}
for(n=0, 25, print1(a(n), ", "))

G.f.: x / Series_Reversion( x*(1 + 9*x)^(1/3) ).
Recurrence: (n-2)*(n-1)*n*a(n) = -216*(2*n - 5)*(4*n - 13)*(4*n - 7)*a(n-3). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ -(-1)^n * 2^(8*n/3 - 13/6) * 3^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 18 2017
G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^5). - Seiichi Manyama, Jun 20 2025

A385205 G.f. A(x) satisfies A(x) = ( 1 + 25xA(x)^4 )^(1/5).

Original entry on oeis.org

1, 5, 50, 500, 4375, 27500, 0, -3562500, -70078125, -876562500, -6926562500, 0, 1189169921875, 25690820312500, 346441406250000, 2911880859375000, 0, -550017993164062500, -12339622131347656250, -171953389892578125000, -1487552714691162109375, 0

Offset: 0

Seiichi Manyama, Jun 21 2025

sign

Cf. A078534, A299958, A385203, A385204.

Cf. A182122, A376636, A385207.

a(n) = 25^n*binomial(4*n/5+1/5, n)/(4*n+1);

a(n) = 25^n * binomial(4*n/5+1/5,n)/(4*n+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^3).
G.f.: ( (1/x) * Series_Reversion(x/(1+25*x)^(4/5)) )^(1/4).
a(5*n+1) = 0 for n > 0.
G.f.: 1/B(x), where B(x) is the g.f. of A299958.

A377269 G.f. A(x) satisfies A(x) = (1 - 9xA(x))^(2/3).

Original entry on oeis.org

1, -6, 27, -90, 189, 0, -1782, 6318, 0, -90882, 360126, 0, -5985819, 24931800, 0, -446371074, 1912892355, 0, -35840971530, 156454458930, 0, -3022929941616, 13367712796110, 0, -264079216747476, 1179032268616902, 0, -23685874363658232, 106533987128598645, 0

Offset: 0

Seiichi Manyama, Oct 22 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A376636, A377268.

A377269[n_] := 9^n*Binomial[(n - 5)/3, n]/(n + 1);
Array[A377269, 35, 0] (* Paolo Xausa, Aug 05 2025 *)

PARI
```
a(n) = 9^n*binomial(n/3-5/3, n)/(n+1);
```

G.f.: (1/x) * Series_Reversion( x/(1-9*x)^(2/3) ).
a(n) = 9^n * binomial(n/3 - 5/3,n)/(n+1).
From Seiichi Manyama, Jun 22 2025: (Start)
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^(1/2)).
a(3*n+2) = 0 for n > 0. (End)
E.g.f.: (27*x^2 + 2*hypergeom([-2/3, 5/6], [1/3, 2/3, 2/3, 1], 4*x^3) - 12*x*hypergeom([-1/3, 7/6], [2/3, 1, 4/3, 4/3], 4*x^3))/2. - Stefano Spezia, Jun 22 2025
D-finite with recurrence n*(n-1)*a(n) -54*(2*n-1)*(n-5)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A385117 G.f. A(x) satisfies A(x) = 1 + 9xA(x)^(2/3).

Original entry on oeis.org

1, 9, 54, 243, 810, 1701, 0, -16038, -56862, 0, 817938, 3241134, 0, -53872371, -224386200, 0, 4017339666, 17216031195, 0, -322568743770, -1408090130370, 0, 27206369474544, 120309415164990, 0, -2376712950727284, -10611290417552118, 0, 213172869272924088

Offset: 0

Seiichi Manyama, Jun 18 2025

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A135864, A214668, A385119.

Cf. A376636.

A385117[n_] := 9^n*Binomial[2*n/3 + 1, n]/(2*n/3 + 1);
Array[A385117, 35, 0] (* Paolo Xausa, Aug 01 2025 *)

a(n) = 9^n*binomial(2*n/3+1, n)/(2*n/3+1);

a(n) = 9^n * binomial(2*n/3+1,n)/(2*n/3+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^(1/3)).
G.f.: 1/B(-x), where B(x) is the g.f. of A135864.
G.f.: B(x)^3, where B(x) is the g.f. of A376636.
a(3*n) = 0 for n > 1.
D-finite with recurrence (n-1)*(n-2)*a(n) + 54*(2*n-3)*(n-6)*a(n-3) = 0. - R. J. Mathar, Jul 30 2025
a(n) ~ A128834(n) * 2^(2*n/3) * 3^(n+3/2) / (sqrt(Pi) * n^(3/2)). - Amiram Eldar, Sep 02 2025

A385207 G.f. A(x) satisfies A(x) = ( 1 + 49xA(x)^6 )^(1/7).

Original entry on oeis.org

1, 7, 147, 3430, 79233, 1714314, 32471124, 450360372, 0, -313409171166, -15459345780879, -537166232508360, -15185812043764453, -348420909370148580, -5588125164812112720, 0, 4783756561471246040577, 254794190560328322173970, 9445124186699596552669050

Offset: 0

Seiichi Manyama, Jun 21 2025

sign

Cf. A385206, A385208.

Cf. A182122, A376636, A385205.

a(n) = 49^n*binomial(6*n/7+1/7, n)/(6*n+1);

a(n) = 49^n * binomial(6*n/7+1/7,n)/(6*n+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^5).
G.f.: ( (1/x) * Series_Reversion(x/(1+49*x)^(6/7)) )^(1/6).
a(7*n+1) = 0 for n > 0.
G.f.: 1/B(-x), where B(x) is the g.f. of A385206.

cp's OEIS Frontend

A004987 a(n) = (3^n/n!)*Product_{k=0..n-1} (3*k + 1). 3-central binomial coefficients.

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A078532 Coefficients of power series that satisfies A(x)^3 - 9*x*A(x)^4 = 1, A(0)=1.

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A245114 G.f. A(x) satisfies A(x)^3 = 1 + 9*x*A(x)^5.

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A008931 Expansion of (2/(1+sqrt(1-36*x)))^(1/3).

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A376282 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^7)^(1/3).

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A247029 G.f. A(x) satisfies A(x) = A(x)^4 - 9*x.

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A004987 a(n) = (3^n/n!)Product_{k=0..n-1} (3k + 1). 3-central binomial coefficients.

A078532 Coefficients of power series that satisfies A(x)^3 - 9xA(x)^4 = 1, A(0)=1.

A245114 G.f. A(x) satisfies A(x)^3 = 1 + 9xA(x)^5.

A376282 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^7)^(1/3).

A385205 G.f. A(x) satisfies A(x) = ( 1 + 25xA(x)^4 )^(1/5).

A377269 G.f. A(x) satisfies A(x) = (1 - 9xA(x))^(2/3).

A385117 G.f. A(x) satisfies A(x) = 1 + 9xA(x)^(2/3).

A385207 G.f. A(x) satisfies A(x) = ( 1 + 49xA(x)^6 )^(1/7).