A004989 -id:A004989 - 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

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

Original entry on oeis.org

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

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

A386415 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^4)^(2/3).

Original entry on oeis.org

1, 6, 135, 4140, 146475, 5629338, 228355281, 9622693080, 417122726490, 18480617374050, 833136935399208, 38094723501749460, 1762459398803643930, 82353342267057244950, 3880848811889775489300, 184228926273804535479216, 8801795826996054546077865, 422898288144162288398536860

Offset: 0

Seiichi Manyama, Jul 21 2025

Paolo Xausa, Table of n, a(n) for n = 0..500
Wikipedia, Fuss-Catalan number

Cf. A004989, A377269, A386413, A386414.

Cf. A078532, A386416.

A386415 := proc(n)
    9^n*binomial((8*n+2)/3,n)/(4*n+1) ;
end proc:
seq(A386415(n),n=0..80) ; # R. J. Mathar, Jul 30 2025

A386415[n_] := 9^n * Binomial[(8*n + 2)/3, n]/(4*n + 1);
Array[A386415, 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, 2/3);

a(n) = 9^n * binomial((8*n+2)/3,n)/(4*n+1).
G.f.: B(x)^2, where B(x) is the g.f. of A386416.
D-finite with recurrence 5*n*(n-1)*(n-2)*(5*n-4)*(5*n+2)*(5*n-7)*(5*n-1)*a(n) -3456*(4*n-11)*(8*n-19)*(8*n-13)*(4*n-5)*(8*n-7)*(2*n-1)*(8*n-1)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A386413 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^2)^(2/3).

Original entry on oeis.org

1, 6, 63, 792, 10935, 160056, 2438667, 38263752, 614014830, 10029572280, 166203389781, 2787232297680, 47213065271268, 806618756189736, 13883029872725475, 240491818267745760, 4189678646994012501, 73357895462268102840, 1290223574267814268290, 22784365638084466567800

Offset: 0

Seiichi Manyama, Jul 21 2025

Paolo Xausa, Table of n, a(n) for n = 0..750
Wikipedia, Fuss-Catalan number

Cf. A004989, A377269, A386414, A386415.

Cf. A078532, A376636.

A386413[n_] := 9^n*Binomial[(4*n + 2)/3, n]/(2*n + 1);
Array[A386413, 25, 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, 4/3, 2/3);

a(n) = 9^n * binomial((4*n+2)/3,n)/(2*n+1).
G.f.: B(x)^2, where B(x) is the g.f. of A078532.
D-finite with recurrence n*(n-2)*(n+2)*a(n) -216*(2*n-5)*(4*n-7)*(4*n-1)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A386414 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^3)^(2/3).

Original entry on oeis.org

1, 6, 99, 2142, 52785, 1404702, 39331656, 1141839504, 34057559052, 1037385419400, 32133013365915, 1009060082062110, 32050934711814915, 1027914968037080970, 33240367148212098900, 1082645830435810233960, 35483717092533680418039, 1169426742892003447650666

Offset: 0

Seiichi Manyama, Jul 21 2025

Paolo Xausa, Table of n, a(n) for n = 0..600
Wikipedia, Fuss-Catalan number

Cf. A004989, A377269, A386413, A386415.

Cf. A004987, A008931.

A386414[n_] := 9^n*Binomial[(6*n + 2)/3, n]/(3*n + 1);
Array[A386414, 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, 2, 2/3);

a(n) = 9^n * binomial((6*n+2)/3,n)/(3*n+1).
G.f.: B(x)^2, where B(x) is the g.f. of A008931.
D-finite with recurrence +n*(3*n+2)*a(n) -6*(6*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Jul 30 2025
G.f.: 2F1(1/3,5/6 ; 5/3 ; 36*x). - R. J. Mathar, Jul 30 2025

A380030 Expansion of e.g.f. (1 - 3xexp(x))^(2/3).

Original entry on oeis.org

1, -2, -6, -26, -208, -2570, -42332, -865718, -21110224, -597416786, -19239912340, -694646155742, -27785653906232, -1219574936748506, -58274685177526300, -3011159013528002150, -167299112903683007392, -9945379044947061586850, -629870278061691615041828

Offset: 0

Seiichi Manyama, Jan 09 2025

Cf. A004989, A380029.

a(n) = n!*sum(k=0, n, (-3)^k*k^(n-k)*binomial(2/3, k)/(n-k)!);

a(n) = n! * Sum_{k=0..n} (-3)^k * k^(n-k) * binomial(2/3,k)/(n-k)!.

cp's OEIS Frontend

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

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

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

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

A386415 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^4)^(2/3).

A386413 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^2)^(2/3).

A386414 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^3)^(2/3).

A380030 Expansion of e.g.f. (1 - 3xexp(x))^(2/3).