A004990 -id:A004990 - cp's OEIS frontend

A004988 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 2).

1, 6, 45, 360, 2970, 24948, 212058, 1817640, 15677145, 135868590, 1182056733, 10316131488, 90266150520, 791564704560, 6954461332920, 61199259729696, 539318476367946, 4758692438540700, 42035116540442850, 371678925199705200, 3289358488017391020

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A004987, A004989, A004990, A004991, A004992, A073006.

List([0..20], n-> 3^n*Product([0..n-1], k-> 3*k+2)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

[1] cat [3^n*&*[3*k+2: k in [0..n-1]]/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 22 2019

A004988 := proc(n)
        binomial(-2/3,n)*(-9)^n ;
end proc: # R. J. Mathar, Sep 16 2012

Table[FullSimplify[9^n*Gamma[n+2/3]/(Gamma[2/3]*Gamma[n+1])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-9x)^(-2/3), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 10 2014 *)
Table[9^n*Pochhammer[2/3, n]/n!, {n,0,20}] (* G. C. Greubel, Aug 22 2019 *)

a(n)=if(n<0,0,prod(k=0,n-1,3*k+2)*3^n/n!)

[9^n*rising_factorial(2/3, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 22 2019

G.f.: (1-9*x)^(-2/3).
a(n) = 9^n*Gegenbauer_C(n,1/3,1). - Paul Barry, Apr 21 2009
a(n) = Product_{k=1..n} (9 - 3/k). - Michel Lagneau, Sep 16 2012
a(n) = (-9)^n*binomial(-2/3, n). - R. J. Mathar, Sep 16 2012
D-finite with recurrence: n*a(n) +3*(-3*n+1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
a(n) = 9^n * Gamma(n+2/3) / (Gamma(2/3) * Gamma(n+1)). - Vaclav Kotesovec, Feb 09 2014
Sum_{n>=0} 1/a(n) = 9/8 + sqrt(3)*Pi/32 - 3*log(3)/32. - Amiram Eldar, Dec 02 2022
Representation as the n-th moment of a positive function on (0, 9): a(n) = Integral_{x = 0..9} x^n * w(x) dx, n >= 0, where w(x) = sqrt(3)/(2*Pi) * 1/(x*(9 - x)^2)^(1/3). The weight function w(x) is the solution of the Hausdorff moment problem on (0, 9) with moments equal to a(n). As a consequence this representation is unique. Cf. A004987. - Peter Bala, Oct 13 2024
a(n) ~ c * 9^n / n^(1/3), where c = 1/Gamma(2/3) = 1/A073006 = 0.738488... . - Amiram Eldar, Aug 17 2025

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

A004989 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k - 2).

Original entry on oeis.org

1, -6, -9, -36, -189, -1134, -7371, -50544, -360126, -2640924, -19806930, -151252920, -1172210130, -9197341020, -72921775230, -583374201840, -4703454502335, -38180983607190, -311811366125385, -2560135427134740, -21121117273861605, -175003543126281870, -1455711290550435555

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004988, A004990, A004991, A004992, A025748, A155579, A185047.

List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k-2)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

[1] cat [3^n*(&*[3*k-2: k in [0..n-1]])/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019

a:= n-> (3^n/n!)*product(3*k-2, k=0..n-1); seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019

Table[9^n*Pochhammer[-2/3, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *)

a(n)=if(n<0,0,prod(k=0,n-1,3*k-2)*3^n/n!)

[9^n*rising_factorial(-2/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019

a(n) ~ -(2/3)*Gamma(1/3)^-1*n^(-5/3)*3^(2*n)*(1 + (5/9)*n^-1 + ...).
G.f.: (1-9*x)^(2/3).
D-finite with recurrence: n*a(n) +3*(-3*n+5)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 99/128 - 5*sqrt(3)*Pi/512 - 15*log(3)/512. - Amiram Eldar, Dec 02 2022
From Peter Bala, Oct 14 2024: (Start)
a(n) = -1/(sqrt(3)*Pi) * 9^n * Gamma(2/3)*Gamma(n-2/3)/Gamma(n+1).
For n >= 1, a(n) = - Integral_{x = 0..9} x^n * w(x) dx, where w(x) = sqrt(3)/(2*Pi) * x^(1/3)*(9 - x)^(2/3)/x^2. (End)

A034164 Related to triple factorial numbers 2*A034000(n+1).

Original entry on oeis.org

1, 5, 30, 198, 1386, 10098, 75735, 580635, 4528953, 35819901, 286559208, 2314516680, 18846778680, 154543585176, 1274984577702, 10574872085646, 88123934047050, 737458184920050, 6194648753328420, 52212039492339540, 441429061162507020, 3742550735942994300

Offset: 0

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A004990, A025748, A034000, A073006, A185047.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 -3*x -(1-9*x)^(1/3))/(3*x)^2 )); // G. C. Greubel, Sep 17 2019

seq(coeff(series((1-3*x-(1-9*x)^(1/3))/(3*x)^2, x, n+2), x, n), n = 0..32); # G. C. Greubel, Sep 17 2019

CoefficientList[Series[ HypergeometricPFQ[{1, 5/3}, {3}, 9 x], {x, 0, 20}], x]
Table[FullSimplify[3^(2*n+1) * Gamma[n+5/3] / ((n+2) * Gamma[2/3] * Gamma[n+2])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1 -3x -(1-9 x)^(1/3))/(3 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)

my(x='x+O('x^30)); Vec((1 -3*x -(1-9*x)^(1/3))/(3*x)^2) \\ G. C. Greubel, Sep 17 2019

def A034164_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1 -3*x -(1-9*x)^(1/3))/(3*x)^2).list()
A034164_list(30) # G. C. Greubel, Sep 17 2019

a(n) = 3^n*(3*n+2)!!!/(n+2)!, where (3*n+2)!!! = 2*A034000(n+1).
G.f.: (1 - 3*x - (1-9*x)^(1/3))/(3*x)^2.
G.f.: 2F1( (1, 5/3); 3; 9 x ). - Olivier Gérard, Feb 15 2011
D-finite with recurrence: (n+2)*a(n) - 3*(3*n+2)*a(n-1) = 0. - R. J. Mathar, Oct 29 2012
a(n) = 3^(2*n+1) * Gamma(n+5/3) / ((n+2) * Gamma(2/3) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014
Integral representation as the n-th moment of a positive function on (0,9): a(n) = Integral_{x=0..9} x^n*W(x) dx, n >= 0, where W(x) = (1/18)*9^(1/3)*sqrt(3)*x^(2/3)*(1-x/9)^(1/3)/Pi. This representation is unique as W(x) is the solution of the Hausdorff moment problem. - Karol A. Penson, Nov 07 2015
Sum_{n>=0} 1/a(n) = 15/16 + (27/64)*(Pi*sqrt(3)/3 - log(3)). - Amiram Eldar, Dec 02 2022
a(n) ~ 3^(2*n+1) * n^(-4/3) / Gamma(2/3). - Amiram Eldar, Aug 19 2025

A248324 Square array read by antidiagonals downwards: super Patalan numbers of order 3.

Original entry on oeis.org

1, 3, 6, 18, 9, 45, 126, 36, 45, 360, 945, 189, 135, 270, 2970, 7371, 1134, 567, 648, 1782, 24948, 58968, 7371, 2835, 2268, 3564, 12474, 212058, 480168, 50544, 15795, 9720, 10692, 21384, 90882, 1817640, 3961386, 360126, 94770, 47385, 40095, 56133, 136323, 681615, 15677145, 33011550, 2640924, 600210, 252720, 173745, 187110, 318087, 908820, 5225715, 135868590

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers of Gessel, A068555, based on Patalan numbers of order 3, A097188.

			T(0..4,0..4) is:
  1    3    18   126   945
  6    9    36   189   1134
  45   45   135  567   2835
  360  270  648  2268  9720
  2970 1782 3564 10692 40095

Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.

Cf. A068555, A248325. First column is A004988, first row is A004987. a(n,1) = -A004990(n+1) = 3*A097188(n). a(1,k) = -A004989(k+1).

T(0,0)=1, T(n,k) = T(n-1,k)*(9*n-3)/(n+k), T(n,k) = T(n,k-1)*(9*k-6)/(n+k).

G.f.: (x/(1-9*x)^(2/3)+y/(1-9*y)^(1/3))/(x+y-9*x*y).

A004991 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 4).

Original entry on oeis.org

1, 12, 126, 1260, 12285, 117936, 1120392, 10563696, 99034650, 924323400, 8596207620, 79710288840, 737320171770, 6806032354800, 62712726697800, 576957085619760, 5300793224131545, 48642573115560060, 445890253559300550, 4083416006279910300, 37363256457461179245, 341606916182502210240

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004988, A004989, A004990, A004992.

List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k+4)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

[1] cat [3^n*(&*[3*k+4: k in [0..n-1]])/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019

a:= n-> (3^n/n!)*product(3*k+4, k=0..n-1); seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019

Table[9^n*Pochhammer[4/3, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *)
Table[3^n/n! Product[3k+4,{k,0,n-1}],{n,0,30}] (* or *) CoefficientList[ Series[ 1/Surd[(1-9x)^4,3],{x,0,30}],x] (* Harvey P. Dale, Aug 02 2021 *)

a(n) = 3^n*prod(k=0,n-1, 3*k+4)/n!;
vector(25, n, n--; a(n)) \\ G. C. Greubel, Aug 22 2019

[9^n*rising_factorial(4/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019

G.f.: (1 - 9*x)^(-4/3).
a(n) ~ 3*Gamma(1/3)^-1*n^(1/3)*3^(2*n)*(1 + 2/9*n^-1 - ...).
a(n) = (3^(2*n))/(Integral_{x=0..1} (1-x^3)^n dx). - Al Hakanson (hawkuu(AT)excite.com), Dec 04 2003
D-finite with recurrence: n*a(n) +3*(-3*n-1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = sqrt(3)*Pi/8 + 3*log(3)/8. - Amiram Eldar, Dec 02 2022

Terms a(16) onward added by G. C. Greubel, Aug 22 2019

A004992 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 5).

Original entry on oeis.org

1, 15, 180, 1980, 20790, 212058, 2120580, 20902860, 203802885, 1970094555, 18912907728, 180532301040, 1715056859880, 16227076443480, 152998149324240, 1438182603647856, 13482961909198650, 126105349621328550, 1176983263132399800, 10964528293391303400, 101970113128539121620

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to the Josephus Problem

Cf. A004988, A004989, A004990, A004991.

List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k+5)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

[1] cat [3^n*&*[3*k+5: 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+5, k=0..n-1): seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019

Table[9^n*Pochhammer[5/3, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *)
Table[3^n/n! Product[3k+5,{k,0,n-1}],{n,0,20}] (* Harvey P. Dale, Jan 07 2023 *)

a(n) = 3^n*prod(k=0,n-1, 3*k+5)/n!;
vector(25, n, a(n-1)) \\ G. C. Greubel, Aug 22 2019

[9^n*rising_factorial(5/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019

G.f.: (1 - 9*x)^(-5/3).
a(n) ~ (3/2)*Gamma(2/3)^-1*n^(2/3)*3^(2*n)*(1 + (5/9)*n^-1 - ...).
D-finite with recurrence: n*a(n) +3*(-3*n-2)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = sqrt(3)*Pi/2 - 3*log(3)/2. - Amiram Eldar, Dec 02 2022

Terms a(16) onward added by G. C. Greubel, Aug 22 2019

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

A301271 Expansion of (1-16*x)^(1/8).

Original entry on oeis.org

1, -2, -14, -140, -1610, -19964, -259532, -3485144, -47920730, -670890220, -9526641124, -136837208872, -1984139528644, -28998962341720, -426699017313880, -6315145456245424, -93937788661650682, -1403541077650545484, -21053116164758182260, -316904801216886322440

Offset: 0

Seiichi Manyama, Jun 15 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..833

Cf. A003557, A007947.

(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), this sequence (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), A303007 (b=240), A303055 (b=504), A305886 (b=1728).

PARI
```
N=20; x='x+O('x^N); Vec((1-16*x)^(1/8))
```

a(n) = 2^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = -sqrt(2-sqrt(2)) * Gamma(1/8) * Gamma(n-1/8) * 16^(n-1) / (Pi*Gamma(n+1)). - Vaclav Kotesovec, Jun 16 2018
a(n) ~ -2^(4*n-3) / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +2*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
a(n) = -2*A097184(n-1). - R. J. Mathar, Jan 20 2020

A303007 Expansion of (1-240*x)^(1/8).

Original entry on oeis.org

1, -30, -3150, -472500, -81506250, -15160162500, -2956231687500, -595469525625000, -122815589660156250, -25791273828632812500, -5493541325498789062500, -1183608449221102734375000, -257434837705589844726562500, -56437637496994696728515625000

Offset: 0

Seiichi Manyama, Jun 15 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..421

Cf. A003557, A007947, A049210, A108091.

(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), A301271 (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), this sequence (b=240), A303055 (b=504), A305886 (b=1728).

CoefficientList[Series[Surd[1-240x,8],{x,0,20}],x] (* Harvey P. Dale, Aug 29 2024 *)

N=20; x='x+O('x^N); Vec((1-240*x)^(1/8))

a(n) = 30^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = 15^n * A301271(n).
a(n) ~ -2^(4*n - 3) * 15^n / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +30*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

cp's OEIS Frontend

A004988 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 2).

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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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A004989 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k - 2).

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A034164 Related to triple factorial numbers 2*A034000(n+1).

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A248324 Square array read by antidiagonals downwards: super Patalan numbers of order 3.

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A004991 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 4).

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A004992 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 5).

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