A034164 -id:A034164 - cp's OEIS frontend

A025748 3rd-order Patalan numbers (generalization of Catalan numbers).

1, 1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620, 1324287183487521060

Offset: 0

Olivier Gérard

G.f. (with a(0)=0) is series reversion of x - 3*x^2 + 3*x^3.

The Hankel transform of a(n) is A005130(n) * 3^binomial(n,2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
I. M. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005, eq. (5.1).
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Apart from the initial 1, identical to A097188.

Cf. A004989, A005130, A034000, A034164, A073006,

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (4 - (1-9*x)^(1/3))/3 )); // G. C. Greubel, Sep 17 2019

A025748 :=proc(n)
        local x;
        coeftayl(4-(1-9*x)^(1/3),x=0,n) ;
        %/3 ;
end proc: # R. J. Mathar, Nov 01 2012

CoefficientList[Series[(4-Power[1-9x, (3)^-1])/3,{x,0,25}],x] (* Harvey P. Dale, Nov 14 2011 *)
Flatten[{1,Table[FullSimplify[9^(n-1) * Gamma[n-1/3] / (n * Gamma[2/3] * Gamma[n])],{n,1,25}]}] (* Vaclav Kotesovec, Feb 09 2014 *)
a[n_] := 9^(n-1) * Pochhammer[2/3, n-1]/n!; a[0] = 1; Array[a, 25, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n)=if(n<1,n==0,polcoeff(serreverse(x-3*x^2+3*x^3+x*O(x^n)),n))

def A025748_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((4 - (1-9*x)^(1/3))/3).list()
A025748_list(25) # G. C. Greubel, Sep 17 2019

From Wolfdieter Lang: (Start)
G.f.: (4 - (1-9*x)^(1/3))/3.
a(n) = 3^(n-1) * 2 * A034000(n-1)/n!, n >= 2.
a(n) = 3 * A034164(n-2), n >= 2. (End)
D-finite with recurrence n*a(n) + 3*(4-3*n)*a(n-1) = 0, n >= 2. - R. J. Mathar, Oct 29 2012
For n>0, a(n) = 9^(n-1) * Gamma(n-1/3) / (n * Gamma(2/3) * Gamma(n)). - Vaclav Kotesovec, Feb 09 2014
For n > 0, a(n) = 3^(2*n-1)*(-1)^(n+1)*binomial(1/3, n). - Peter Bala, Mar 01 2022
Sum_{n>=0} 1/a(n) = 37/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022
For n >= 1, a(n) = Integral_{x = 0..9} x^n * w(x) dx, where w(x) = 1/(2*sqrt(3)*Pi) *  x^(2/3)*(9 - x)^(1/3)/x^2. - Peter Bala, Oct 14 2024
a(n) ~ 9^(n-1) / (Gamma(2/3) * n^(4/3)). - Amiram Eldar, Aug 20 2025

A004990 a(n) = (3^n/n!)Product_{k=0..n-1} (3k - 1).

Original entry on oeis.org

1, -3, -9, -45, -270, -1782, -12474, -90882, -681615, -5225715, -40760577, -322379109, -2579032872, -20830650120, -169621008120, -1390892266584, -11474861199318, -95173848770814, -793115406423450, -6637123664280450, -55751838779955780

Offset: 0

Joe Keane (jgk(AT)jgk.org)

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

Cf. A004988, A004989, A004991, A004992, A034164.

List([0..20], 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..20]]; // G. C. Greubel, Aug 22 2019

a:= n-> (3^n/n!)*mul(3*k-1, k=0..n-1): seq(a(n), n=0..20); # G. C. Greubel, Aug 22 2019

FullSimplify[Table[3^(2*n) * Gamma[n-1/3] / (n! * Gamma[-1/3]),{n,0,20}]] (* Vaclav Kotesovec, Dec 03 2014 *)

for(n=0,30,print1( (3^n/n!)*prod(k=0,n-1,(3*k-1) ),","))

[9^n*rising_factorial(-1/3, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 22 2019

a(n) = 9*A034164(n-2), n >= 2.
G.f.: (1 - 9*x)^(1/3).
a(n) ~ -1/3*Gamma(2/3)^-1*n^(-4/3)*3^(2*n)*{1 + 2/9*n^-1 + ...}.
G.f.: 1 + 3*x/(G(0)-3*x) where G(k) = (1+9*x)*k + 1 - 3*x - 3*x*(k+1)*(3*k+2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 07 2012
D-finite with recurrence: n*a(n) +3*(-3*n+4)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 9/16 - sqrt(3)*Pi/64 + 3*log(3)/64. - Amiram Eldar, Dec 02 2022
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-9)^n*binomial(1/3, n).
E.g.f.: hypergeom([-1/3], [1], 9*x).
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) = A004989.
Sum_{k = 0..2*n} a(k)*a(2*n-k) = 18^n/(2*n)! * Product_{k = 1..n} (6*k - 5)*(3*k - 4). (End)

More terms from Jason Earls, Dec 03 2001

A097188 G.f. A(x) satisfies A057083(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097186.

Original entry on oeis.org

1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620

Offset: 0

Paul D. Hanna, Aug 03 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4, eq.(23) for l=4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Cf. A004988, A025748, A034164, A057083, A097186.

Essentially identical to A025748.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1 - (1-9*x)^(1/3))/(3*x) )); // G. C. Greubel, Sep 17 2019

seq(coeff(series((1-(1-9*x)^(1/3))/(3*x), x, n+2), x, n), n = 0..25); # G. C. Greubel, Sep 17 2019

Table[FullSimplify[9^n * Gamma[n+2/3] / ((n+1) * Gamma[2/3] * Gamma[n+1])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-(1 - 9 x)^(1/3))/(3 x), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 10 2014 *)

a(n)=polcoeff((1-(1-9*x+x^2*O(x^n))^(1/3))/(3*x),n,x)

def A097188_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1 - (1-9*x)^(1/3))/(3*x)).list()
A097188_list(25) # G. C. Greubel, Sep 17 2019

G.f.: A(x) = (1 - (1-9*x)^(1/3))/(3*x).
G.f.: A(x) = (1/x)*(series reversion of x/A057083(x)).
a(n) = A004988(n)/(n+1).
a(n) = A025748(n+1).
a(n) = 3*A034164(n-1) for n>=1.
x*A(x) is the compositional inverse of x-3*x^2+3*x^3. - Ira M. Gessel, Feb 18 2012
a(n) = 1/(n+1) * Sum_{k=1..n} binomial(k,n-k) * 3^(k)*(-1)^(n-k) * binomial(n+k,n), if n>0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011
Conjecture: (n+1)*a(n) +3*(-3*n+1)*a(n-1)=0. - R. J. Mathar, Nov 16 2012
a(n) = 9^n * Gamma(n+2/3) / ((n+1) * Gamma(2/3) * Gamma(n+1)). - Vaclav Kotesovec, Feb 09 2014
Sum_{n>=0} 1/a(n) = 21/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022

A034171 Related to triple factorial numbers A007559(n+1).

Original entry on oeis.org

1, 6, 42, 315, 2457, 19656, 160056, 1320462, 11003850, 92432340, 781473420, 6642524070, 56716936290, 486145168200, 4180848446520, 36059817851235, 311811366125385, 2702365173086670, 23467908082068450, 204170800313995515, 1779202688450532345, 15527587099204645920

Offset: 0

Wolfdieter Lang

Working with an offset of 1, we conjecture a(p*n) = a(n) (mod p^2) for prime p = 1 (mod 3) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/3 and 1 <= k <= (p-1)/3. Cf. A298799, A004981 and A004982. - Peter Bala, Dec 23 2019

Michael De Vlieger, Table of n, a(n) for n = 0..1050
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A007559, A025748, A034164, A035529.

Cf. A298799, A004981, A004982, A004987, A073005.

CoefficientList[Series[(-1 + (1 - 9 x)^(-1/3))/(3 x), {x, 0, 19}], x] (* Michael De Vlieger, Oct 13 2019 *)

a(n) = 3^n*A007559(n+1)/(n+1)! where A007559(n+1)=(3*n+1)!!!.
G.f.: (-1+(1-9*x)^(-1/3))/(3*x).
a(n) = A035529(n+1, 1) (first column of triangle).
Convolution of A004987(n) with A025748(n+1), n >= 0.
From R. J. Mathar, Jan 28 2020: (Start)
D-finite with recurrence: (n+1)*a(n) + 3*(-3*n-1)*a(n-1) = 0.
G.f.: (1F0(1/3;;9*x)-1)/(3*x). (End)
Sum_{n>=0} 1/a(n) = 3/8 + 3*sqrt(3)*Pi/32 + 9*log(3)/32. - Amiram Eldar, Dec 22 2022
a(n) ~ 3^(2*n+1) * n^(-2/3) / Gamma(1/3). - Amiram Eldar, Aug 19 2025

A067622 Consider the power series (x + 1)^(1/3) = 1 + x/3-x^2/9 + 5x^3/81 + ...; sequence gives numerators of coefficients.

Original entry on oeis.org

1, 1, -1, 5, -10, 22, -154, 374, -935, 21505, -55913, 147407, -1179256, 3174920, -8617640, 70664648, -194327782, 537259162, -13431479050, 37466757350, -104906920580, 884215473460, -2491879970660, 7042269482300, -59859290599550

Offset: 0

Benoit Cloitre, Feb 02 2002

a(n) is also the numerator of the binomial coefficient C(k,n) evaluated at k=1/3, e.g. a(4) = (1/24)k(k-1)(k-2)(k-3), plug in k=1/3 and take numerator. - James R. Buddenhagen, Aug 16 2014

Denominators are A067623.

s := convert(taylor((x+1)^(1/3), x, 50), polynom): for n from 0 to 50 do printf(`%a,`, abs(numer(coeff(s, x, n)))) od;
seq(numer(subs(k=1/3,expand(binomial(k,n)))),n=0..50) # James R. Buddenhagen, Aug 16 2014

a(n) =(-1)^n*A004990(n)*A067623(n)/A000244(n); ignoring signs, a(n) =A038502(A004990(n)) =A038502(A034164(n-2)). a(n)'s sign is (-1)^(n+1) if n>0.

Edited by Henry Bottomley and James Sellers, Feb 11 2002

A185047 Expansion of 2F1( [1, 4/3]; [3]; 9*x).

Original entry on oeis.org

1, 4, 21, 126, 819, 5616, 40014, 293436, 2200770, 16805880, 130245570, 1021926780, 8102419470, 64819355760, 522606055815, 4242331511910, 34645707347265, 284459491903860, 2346790808206845, 19444838125142430, 161745698950048395, 1350224965148230080

Offset: 0

Olivier Gérard, Feb 15 2011

Close to A003168.

Can be seen as a degree 3 analog of the Catalan numbers A000108 (which would be degree 2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A003168, A034164.

A185047:=n->-9^(n+1)*binomial(n+1/3, n+2): seq(A185047(n), n=0..30); # Wesley Ivan Hurt, Feb 16 2017

CoefficientList[Series[ HypergeometricPFQ[{1, 4/3}, {3}, 9 x], {x, 0, 20}], x]

a(n) = -9^(n+1)*binomial(n+1/3, n+2). - Karol A. Penson, Nov 06 2015
a(n) = (1/(6*sqrt(3)*Pi))*Integral_{x = 0..9} x^n*x^(1/3)*(9 - x)^(2/3). Cf. A034164. - Peter Bala, Nov 17 2015
O.g.f.: (1 - (1-9*x)^(2/3) - 6*x)/(9*x^2).
D-finite with recurrence (n+2)*a(n) +3*(-3*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
Sum_{n>=0} 1/a(n) = (3/512)*(92 + 15*Pi*sqrt(3) + 45*log(3)). - Amiram Eldar, Dec 18 2022
a(n) = ((n + 3)/3) * Product_{1 <= i <= j <= n} (2*i + j + 3)/(2*i + j - 1). - Peter Bala, Feb 22 2023

cp's OEIS Frontend

A025748 3rd-order Patalan numbers (generalization of Catalan numbers).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A004990 a(n) = (3^n/n!)*Product_{k=0..n-1} (3*k - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A097188 G.f. A(x) satisfies A057083(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097186.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A034171 Related to triple factorial numbers A007559(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A067622 Consider the power series (x + 1)^(1/3) = 1 + x/3-x^2/9 + 5x^3/81 + ...; sequence gives numerators of coefficients.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

Extensions

A185047 Expansion of 2F1( [1, 4/3]; [3]; 9*x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A004990 a(n) = (3^n/n!)Product_{k=0..n-1} (3k - 1).