A214668 -id:A214668 - cp's OEIS frontend

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

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

A135864 G.f. A(x) satisfies A(x) = 1 + 9xA(x)^(1/3).

Original entry on oeis.org

1, 9, 27, 0, -81, 243, 0, -2916, 10935, 0, -168399, 682344, 0, -11711385, 49364964, 0, -900438201, 3887490915, 0, -73729998576, 323480165085, 0, -6303914878248, 27978933759300, 0, -556251967191492, 2490404893915293, 0, -50276620111538700, 226626845346291663, 0

Offset: 0

Paul D. Hanna, Dec 02 2007

sign

Cf. A135863, A135866, A214668.

PARI
```
a(n)=9^n*binomial(n/3,n)/(1-2*n/3)
```

a(n) = -9^n*binomial(n/3,n)/(2n/3 - 1).

G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^(-1/3)). - Seiichi Manyama, Jun 18 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

A385119 G.f. A(x) satisfies A(x) = 1 + 9xA(x)^(5/3).

Original entry on oeis.org

1, 9, 135, 2430, 48195, 1015740, 22320522, 505692720, 11727186075, 277005649635, 6641224015140, 161193712078854, 3953072078945730, 97801207953712200, 2438092322304120720, 61182608813245896840, 1544295394480280288715, 39180450803555268621540

Offset: 0

Seiichi Manyama, Jun 18 2025

Paolo Xausa, Table of n, a(n) for n = 0..650

Cf. A135864, A214668, A385117.

Cf. A245114.

A385119[n_] := 9^n*Binomial[#, n]/# & [5*n/3 + 1];
Array[A385119, 20, 0] (* Paolo Xausa, Aug 05 2025 *)

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

a(n) = 9^n * binomial(5*n/3+1,n)/(5*n/3+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^(7/3)).
G.f.: B(x)^3, where B(x) is the g.f. of A245114.
D-finite with recurrence 2*n*(n-1)*(n-2)*(2*n+3)*a(n) - 135*(5*n-9)*(5*n-3)*(5*n-12)*(5*n-6)*a(n-3) = 0. - R. J. Mathar, Jul 30 2025
a(n) ~ 3^(n+1) * 5^(5*n/3+1/2) / (sqrt(Pi) * 2^(2*(n+3)/3) * n^(3/2)). - Amiram Eldar, Sep 02 2025

cp's OEIS Frontend

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

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A135864 G.f. A(x) satisfies A(x) = 1 + 9*x*A(x)^(1/3).

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

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A385119 G.f. A(x) satisfies A(x) = 1 + 9*x*A(x)^(5/3).

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

A135864 G.f. A(x) satisfies A(x) = 1 + 9xA(x)^(1/3).

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

A385119 G.f. A(x) satisfies A(x) = 1 + 9xA(x)^(5/3).