A376282 -id:A376282 - 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

A376636 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^2)^(1/3).

Original entry on oeis.org

1, 3, 9, 18, 0, -162, -567, 0, 8019, 31590, 0, -520506, -2160756, 0, 38480265, 164549880, 0, -3072083274, -13390246485, 0, 258054995016, 1139882486490, 0, -22474826957232, -100257845970825, 0, 2011064804461548, 9039247392729582, 0, -183769714890451800

Offset: 0

Seiichi Manyama, Oct 23 2024

sign

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

Cf. A004987, A008931, A078532, A245114, A247029, A376282.

Cf. A385117.

A376636[n_] := 9^n*Binomial[(2*n + 1)/3, n]/(2*n + 1);
Array[A376636, 35, 0] (* Paolo Xausa, Aug 04 2025 *)

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

a(n) = 9^n * binomial(2*n/3 + 1/3,n)/(2*n+1).
From Seiichi Manyama, Jun 20 2025: (Start)
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)).
a(3*n+1) = 0 for n > 0. (End)
D-finite with recurrence n*(n-2)*a(n) +54*(2*n-5)*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

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

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

A386416 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^8)^(1/3).

Original entry on oeis.org

1, 3, 63, 1881, 65610, 2499336, 100777122, 4228144596, 182674383705, 8072369224920, 363154406671485, 16576444298006658, 765806677899249168, 35739548618003938440, 1682429522012566325460, 79793991407758199002740, 3809208342822290233767522, 182890356905449116974950200

Offset: 0

Seiichi Manyama, Jul 21 2025

Paolo Xausa, Table of n, a(n) for n = 0..550
Wikipedia, Fuss-Catalan number

Cf. A004990, A377268, A376636, A004987, A078532, A245114, A008931, A376282.

Cf. A386415.

A386416[n_] := 9^n*Binomial[(8*n + 1)/3, n]/(8*n + 1);
Array[A386416, 20, 0] (* Paolo Xausa, Aug 01 2025 *)

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = 9^n*apr(n, 8/3, 1/3);

a(n) = 9^n * binomial((8*n+1)/3,n)/(8*n+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^13).
D-finite with recurrence 5*n*(n-1)*(n-2)*(5*n-8)*(5*n-11)*(5*n+1)*(5*n-2)*a(n) -3456*(8*n-11)*(8*n-5)*(4*n-1)*(8*n-23)*(2*n-5)*(8*n-17)*(4*n-7)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

PARI

Sage

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A376636 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^2)^(1/3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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.

A376636 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^2)^(1/3).

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

A386416 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^8)^(1/3).