A034000 -id:A034000 - cp's OEIS frontend

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

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

A008544 Triple factorial numbers: Product_{k=0..n-1} (3*k+2).

Original entry on oeis.org

1, 2, 10, 80, 880, 12320, 209440, 4188800, 96342400, 2504902400, 72642169600, 2324549427200, 81359229952000, 3091650738176000, 126757680265216000, 5577337931669504000, 262134882788466688000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

a(n-1), n >= 1, enumerates increasing plane (aka ordered) trees with n vertices (one of them a root labeled 1) where each vertex with outdegree r >= 0 comes in r+1 types (like an (r+1)-ary vertex). See the increasing tree comments under A004747. - Wolfdieter Lang, Oct 12 2007
An example for the case of 3 vertices is shown below. For the enumeration of non-plane trees of this type see A029768. - Peter Bala, Aug 30 2011
a(n) is the product of the positive integers k <= 3*n that have k modulo 3 = 2. - Peter Luschny, Jun 23 2011
See A094638 for connections to differential operators. - Tom Copeland, Sep 20 2011
Partial products of A016789. - Reinhard Zumkeller, Sep 20 2013
The Mathar conjecture is true. Generally from the factorial form, the last term is the "extra" product beyond the prior term, from k=n-1 and 3k+2 evaluates to 3*(n-1)+2 = 3n-1, yielding a(n) = a(n-1)*(3n-1) (eqn1). Similarly, a(n) = a(n-2)*(3n-1)*(3(n-2)+2) = a(n-2)*(3n-1)*(3n-4) (eqn2) and a(n) = a(n-3)*(3n-1)*(3n-4)*(3*(n-2)+2) = a(n-3)*(3n-1)*(3n-4)*(3n-7) (eqn3). We equate (eqn2) and (eqn3) to get a(n-2)*(3n-1)*(3n-4) = a(n-3)*(3n-1)*(3n-4)*(3n-7) or a(n-2)+(7-3n)*a(n-3) = 0 (eqn4). From (eqn1) we have a(n)+(1-3n)*a(n-1) = 0 (eqn5). Combining (eqn4) and (eqn5) yields a(n)+(1-3n)*a(n-1)+a(n-2)+(7-3n)*a(n-3) = 0. - Bill McEachen, Jan 01 2016
a(n-1), n>=1, is the dimension of the n-th component of the operad encoding the multilinearization of the following identity in nonassociative algebras: s*(a,a,b)-(s+t)*(a,b,a)+t*(b,a,a)=0, for any given pair of scalars (s,t). Here (a,b,c) is the associator (ab)c-a(bc). This is proved in the referenced article on associator dependent algebras by Bremner and me. - Vladimir Dotsenko, Mar 22 2022

			a(2) = 10 from the described trees with 3 vertices: there are three trees with a root vertex (label 1) with outdegree r=2 (like the three 3-stars each with one different ray missing) and the four trees with a root (r=1 and label 1) a vertex with (r=1) and a leaf (r=0). Assigning labels 2 and 3 yields 2*3+4=10 such trees.
a(2) = 10. The 10 possible plane increasing trees on 3 vertices, where vertices of outdegree 1 come in 2 colors (denoted a or b) and vertices of outdegree 2 come in 3 colors (a, b or c), are:
.
   1a    1b    1a    1b        1a       1b       1c
   |     |     |     |        / \      / \      / \
   2a    2b    2b    2a      2   3    2   3    2   3
   |     |     |     |
   3     3     3     3         1a       1b       1c
                              / \      / \      / \
                             3   2    3   2    3   2

T. D. Noe, Table of n, a(n) for n = 0..100
Murray Bremner and Vladimir Dotsenko, Associator dependent algebras and Koszul duality, arXiv:2203.11142 [math.RA], 2022.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Keiichi Shigechi, On the lattice of weighted partitions, arXiv:2212.14666 [math.CO], 2022. See p. 27.
Garrett Southwood and Hua Wang, The Statistics and Combinatorics of Increasing Trees and Colored Increasing Trees, J. Combin. Math. Combin. Comput. (2024) Vol. 123, 475-487. See p. 485.

a(n) = A004747(n+1, 1) (first column of triangle). Cf. A051141.
Cf. A000165, A001813, A047055, A047657, A084947, A084948, A084949.
Cf. A029768, A034724, A049308.
Cf. A024396, A193534.
Cf. A225470, A290596 (first columns).
Subsequence of A007661.

a008544 n = a008544_list !! n
a008544_list = scanl (*) 1 a016789_list
-- Reinhard Zumkeller, Sep 20 2013

[Round((Gamma(2*n-5/3)/Gamma(n-5/6)*Gamma(2/3)/Gamma(5/6) )/ Sqrt(3)*3^n/4^(n-1)): n in [1..20]]; // Vincenzo Librandi, Feb 21 2015

[Round(3^n*Gamma(n+2/3)/Gamma(2/3)): n in [0..20]]; // G. C. Greubel, Mar 31 2019

a := n -> mul(3*k-1, k = 1..n);
A008544 := n -> mul(k, k = select(k-> k mod 3 = 2, [$1 .. 3*n])): seq(A008544(n), n = 0 .. 16); # Peter Luschny, Jun 23 2011

k = 3; b[1]=2; b[n_]:= b[n] = b[n-1]+k; a[0]=1; a[1]=2; a[n_]:= a[n] = a[n-1]*b[n]; Table[a[n], {n,0,20}] (* Roger L. Bagula, Sep 17 2008 *)
Product[3 k + 2, {k, 0, # - 1}] & /@ Range[0, 16] (* Michael De Vlieger, Jan 02 2016 *)
Table[3^n*Pochhammer[2/3, n], {n,0,20}] (* G. C. Greubel, Mar 31 2019 *)

a(n):=((n)!*sum(binomial(k,n-k)*binomial(n+k,k)*3^(-n+k)*(-1)^(n-k),k,floor(n/2),n)); /* Vladimir Kruchinin, Sep 28 2013 */

PARI
```
a(n) = prod(k=0,n-1, 3*k+2 );
```

vector(20, n, n--; round(3^n*gamma(n+2/3)/gamma(2/3))) \\ G. C. Greubel, Mar 31 2019

@CachedFunction
def A008544(n): return 1 if n == 0 else (3*n-1)*A008544(n-1)
[A008544(n) for n in (0..16)]  # Peter Luschny, May 20 2013

[3^n*rising_factorial(2/3, n) for n in (0..20)] # G. C. Greubel, Mar 31 2019

a(n) = Product_{k=0..n-1} (3*k+2) = A007661(3*n-1) (with A007661(-1) = 1).
E.g.f.: (1-3*x)^(-2/3).
a(n) = 2*A034000(n), n >= 1, a(0) = 1.
a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(2/3)^-1*n^(1/6)*3^n*e^-n*n^n*{1 - 1/36*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
a(n) = (Gamma(2*n-5/3)/Gamma(n-5/6)*Gamma(2/3)/Gamma(5/6))/sqrt(3)*3^n/4^(n-1). - Jeremy L. Martin, Mar 31 2002 (typo fixed by Vincenzo Librandi, Feb 21 2015)
From Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003: (Start)
a(n) = A084939(n)/A000142(n)*A000079(n).
a(n) = 3^n*Pochhammer(2/3, n) = 3^n*Gamma(n+2/3)/Gamma(2/3). (End)
Let T = A094638 and c(t) = column vector(1, t, t^2, t^3, t^4, t^5,...), then A008544 = unsigned [ T * c(-3) ] and the list partition transform A133314 of [1,T * c(-3)] gives [1,T * c(3)] with all odd terms negated, which equals a signed version of A007559; i.e., LPT[(1,signed A008544)] = signed A007559. Also LPT[A007559] = (1,-A008544) and e.g.f. [1,T * c(t)] = (1-x*t)^(-1/t) for t = 3 or -3. Analogous results hold for the double factorial, quadruple factorial and so on. - Tom Copeland, Dec 22 2007
G.f.: 1/(1-2x/(1-3x/(1-5x/(1-6x/(1-8x/(1-9x/(1-11x/(1-12x/(1-...))))))))) (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k=0..n} 3^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0) where Q(k) = 1 - x*(3*k+2)/(1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+2)/(x*(3*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
D-finite with recurrence: a(n) = (9*(n-2)*(n-1)+2)*a(n-2) + 4*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 09 2013
a(n) = n!*Sum_{k=floor(n/2)..n} binomial(k,n-k)*binomial(n+k,k)*3^(-n+k)*(-1)^(n-k). - Vladimir Kruchinin, Sep 28 2013
Recurrence equation: a(n) = 3*a(n-1) + (3*n - 4)^2*a(n-2) with a(0) = 1 and a(1) = 2. A024396 satisfies the same recurrence (but with different initial conditions). This observation leads to a continued fraction expansion for the constant A193534 due to Euler. - Peter Bala, Feb 20 2015
a(n) = A225470(n, 0), n >= 0. - Wolfdieter Lang, May 29 2017
G.f.: Hypergeometric2F0(1, 2/3; -; 3*x). - G. C. Greubel, Mar 31 2019
D-finite with recurrence: a(n) + (-3*n+1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
G.f.: 1/(1-2*x-6*x^2/(1-8*x-30*x^2/(1-14*x-72*x^2/(1-20*x-132*x^2/(1-...))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Feb 28 2020
G.f.: 1/G(0), where G(k) = 1 - (6*k+2)*x - 3*(k+1)*(3*k+2)*x^2/G(k+1). - Nikolaos Pantelidis, Feb 28 2020
Sum_{n>=0} 1/a(n) = 1 + (e/3)^(1/3) * (Gamma(2/3) - Gamma(2/3, 1/3)). - Amiram Eldar, Mar 01 2022

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

A034001 One third of triple factorial numbers.

Original entry on oeis.org

1, 6, 54, 648, 9720, 174960, 3674160, 88179840, 2380855680, 71425670400, 2357047123200, 84853696435200, 3309294160972800, 138990354760857600, 6254565964238592000, 300219166283452416000, 15311177480456073216000

Offset: 1

Wolfdieter Lang

Muniru A Asiru, Table of n, a(n) for n = 1..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 495.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order, Vol. 21 (2004), pp. 83-89; arXiv preprint, arXiv:math/0307064 [math.CO], 2003.

Cf. A007559, A034000.

List([1..20],n->3^(n-1)*Factorial(n)); # Muniru A Asiru, Jul 28 2018

[3^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 15 2019

G(x):=(1-3*x)^(n-3): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=0..16); # Zerinvary Lajos, Apr 04 2009

terms = 17;
CoefficientList[1/(1-3x)^2 + O[x]^terms, x] Range[0, terms-1]! (* Jean-François Alcover, Jul 28 2018 *)
Table[3^(n-1)*n!, {n,20}] (* G. C. Greubel, Aug 15 2019 *)

vector(20, n, 3^(n-1)*n!) \\ G. C. Greubel, Aug 15 2019

[3^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019

3*a(n) = (3*n)!!! = Product_{j=1..n} 3*j = 3^n*n!.
E.g.f.: (-1 + 1/(1-3*x))/3.
E.g.f.: 1/(1-3*x)^2. - Paul Barry, Sep 14 2004. For offset 0. - Wolfdieter Lang, Apr 06 2017
D-finite with recurrence a(n) - 3*n*a(n-1) = 0. - R. J. Mathar, Dec 02 2012
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*(exp(1/3)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(1-exp(-1/3)). (End)

A035022 One eighth of 9-factorial numbers.

Original entry on oeis.org

1, 17, 442, 15470, 680680, 36076040, 2236714480, 158806728080, 12704538246400, 1130703903929600, 110808982585100800, 11856561136605785600, 1375361091846271129600, 171920136480783891200000, 23037298288425041420800000, 3294333655244780923174400000, 500738715597206700322508800000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A007559, A034000, A034171, A035012, A035013, A035017, A049211.

Cf. A035018, A035020, A035021, A035022, A035023, A045756.

[n le 1 select 1 else (9*n-1)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 19 2022

f := gfun:-rectoproc({(9*n - 1)*a(n - 1) - a(n) = 0, a(1) = 1}, a(n), remember);
map(f, [$ (1 .. 20)]); # Georg Fischer, Feb 15 2020

Table[9^n*Pochhammer[8/9, n]/8, {n, 40}] (* G. C. Greubel, Oct 19 2022 *)

[9^n*rising_factorial(8/9,n)/8 for n in range(1,40)] # G. C. Greubel, Oct 19 2022

8*a(n) = (9*n-1)(!^9) := Product_{j=1..n} (9*j - 1).
a(n) = (9*n)!/(n!*2^4*3^(4*n)*5*7*A045756(n)*A035012(n)*A007559(n)*A035017(n) *A035018(n)*A034000(n) *A035021(n)).
E.g.f.: (-1+(1-9*x)^(-8/9))/8.
D-finite with recurrence: a(1) = 1, a(n) = (9*n - 1)*a(n-1) for n > 1. - Georg Fischer, Feb 15 2020
a(n) = (1/8) * 9^n * Pochhammer(n, 8/9). - G. C. Greubel, Oct 19 2022
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A049211(n)/8.
Sum_{n>=1} 1/a(n) = 8*(e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). (End)

A035020 One sixth of 9-factorial numbers.

Original entry on oeis.org

1, 15, 360, 11880, 498960, 25446960, 1526817600, 105350414400, 8217332323200, 714907912118400, 68631159563366400, 7206271754153472000, 821514979973495808000, 101046342536739984384000, 13338117214849677938688000, 1880674527293804589355008000, 282101179094070688403251200000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A007559, A034000, A034171, A035012, A035013, A035017, A147630.

Cf. A035018, A035020, A035021, A035022, A035023, A045756.

[n le 1 select 1 else (9*n-3)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 14, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[2/3, n]/6, {n, 40}] (* G. C. Greubel, Oct 18 2022 *)

[9^n*rising_factorial(2/3,n)/6 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

6*a(n) = (9*n-3)(!^9) := Product_{j=1..n} (9*j-3) = 3^n*2*A034000(n), where 2*A034000(n) = (3*n-1)(!^3) := Product_{j=1..n} (3*j-1).
E.g.f.: (-1+(1-9*x)^(-2/3))/6.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/6) * 9^n * Pochhammer(n, 2/3).
a(n) = (9*n - 3)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147630(n+1)/6.
Sum_{n>=1} 1/a(n) = 6*(e/9^3)^(1/9)*(Gamma(2/3) - Gamma(2/3, 1/9)). (End)

A034724 a(n) = n-th sextic factorial number divided by 4.

Original entry on oeis.org

1, 10, 160, 3520, 98560, 3351040, 134041600, 6165913600, 320627507200, 18596395417600, 1190169306726400, 83311851470848000, 6331700711784448000, 519199458366324736000, 45689552336236576768000, 4294817919606238216192000, 429481791960623821619200000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..345

Cf. A034000, A008542, A034689, A073006.

List([1..20], n-> Product([1..n], j-> 6*j-2)/4 ); # G. C. Greubel, Nov 11 2019

[(&*[6*j-2: j in [1..n]])/4: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(6*j-2, j=1..n)/4, n=1..20); # G. C. Greubel, Nov 11 2019

With[{nn=20},CoefficientList[Series[((1-6x)^(-2/3)-1)/4,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 02 2017 *)
Table[6^n*Pochhammer[2/3, n]/4, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 6*j-2)/4 ) \\ G. C. Greubel, Nov 11 2019

[product( (6*j-2) for j in (1..n))/4 for n in (1..20)] # G. C. Greubel, Nov 11 2019

4*a(n) = (6*n-2)(!^6) = Product_{j=1..n} (6*j-2).
a(n) = 2^(n+1)*A034000(n), 2*A034000(n) = (3*n-1)(!^3).
E.g.f.: (-1 + (1-6*x)^(-2/3))/4.
D-finite with recurrence: a(n) +2*(-3*n+1)*a(n-1)=0. - R. J. Mathar, Jan 28 2020
Sum_{n>=1} 1/a(n) = 4*(e/6^2)^(1/6)*(Gamma(2/3) - Gamma(2/3, 1/6)). - Amiram Eldar, Dec 18 2022
a(n) ~ sqrt(Pi) * 2^(n-3/2) * (3/e)^n * n^(n+1/6) / Gamma(2/3). - Amiram Eldar, Sep 01 2025

A034787 a(n) = n-th sextic factorial number divided by 5.

Original entry on oeis.org

1, 11, 187, 4301, 124729, 4365515, 178986115, 8412347405, 445854412465, 26305410335435, 1709851671803275, 121399468698032525, 9347759089748504425, 775864004449125867275, 69051896395972202187475, 6559930157617359207810125, 662552945919353279988822625

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..345

Cf. A008542, A034689, A034723, A034724, A034788.

List([1..20], n-> Product([1..n], j-> 6*j-1)/5 ); # G. C. Greubel, Nov 11 2019

[(&*[6*j-1: j in [1..n]])/5: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(6*j-1, j=1..n)/5, n=1..20); # G. C. Greubel, Nov 11 2019

Table[6^n*Pochhammer[5/6, n]/5, {n,20}] (* G. C. Greubel, Nov 11 2019 *)
With[{nn=20},CoefficientList[Series[(-1+(1-6x)^(-5/6))/5,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 21 2024 *)

vector(20, n, prod(j=1,n, 6*j-1)/5 ) \\ G. C. Greubel, Nov 11 2019

[product( (6*j-1) for j in (1..n))/5 for n in (1..20)] # G. C. Greubel, Nov 11 2019

5*a(n) = (6*n-1)(!^6) = Product_{j=1..n} (6*j-1) = (6*n)!/(3^(2*n)*2^(2*n+1)*(2*n)!*A008542(n)*A007559(n)*A034000(n)).
E.g.f.: (-1 + (1-6*x)^(-5/6))/5.
a(n+1) ~ sqrt(2*Pi) * 6/(5*Gamma(5/6)) * n^(4/3) * (6*n/e)^n * (1 + (61/72)/n + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
D-finite with recurrence: a(n) +(-6*n+1)*a(n-1)=0. - R. J. Mathar, Feb 24 2020
Sum_{n>=1} 1/a(n) = 5*(e/6)^(1/6)*(Gamma(5/6) - Gamma(5/6, 1/6)). - Amiram Eldar, Dec 18 2022

A051604 a(n) = (3*n+4)!!!/4!!!.

Original entry on oeis.org

1, 7, 70, 910, 14560, 276640, 6086080, 152152000, 4260256000, 132067936000, 4490309824000, 166141463488000, 6645658539520000, 285763317199360000, 13145112591170560000, 644110516967357440000, 33493746882302586880000, 1842156078526642278400000

Offset: 0

Wolfdieter Lang

Related to A007559(n+1) ((3*n+1)!!! triple factorials).

Row m=4 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..378

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1). (rows m=0..3).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(7/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

With[{nn = 30}, CoefficientList[Series[1/(1-3*x)^(7/3), {x, 0, nn}], x]* Range[0,nn]! ] (* G. C. Greubel, Aug 15 2018 *)
With[{c=Times@@Range[4,1,-3]},Table[(Times@@Range[3n+4,1,-3])/c,{n,0,20}]] (* Harvey P. Dale, Feb 06 2023 *)

x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(7/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((3*n+4)(!^3))/4(!^3).
E.g.f.: 1/(1-3*x)^(7/3).
Sum_{n>=0} 1/a(n) = 1 + 3*(3*e)^(1/3)*(Gamma(7/3) - Gamma(7/3, 1/3)). - Amiram Eldar, Dec 23 2022

A051605 a(n) = (3*n+5)!!!/5!!!.

Original entry on oeis.org

1, 8, 88, 1232, 20944, 418880, 9634240, 250490240, 7264216960, 232454942720, 8135922995200, 309165073817600, 12675768026521600, 557733793166950400, 26213488278846668800, 1310674413942333440000, 69465743938943672320000, 3890081660580845649920000

Offset: 0

Wolfdieter Lang

Related to A008544(n+1) ((3*n+2)!!! triple factorials).

Row m=5 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..378

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604 (rows m=0..4).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(8/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

RecurrenceTable[{a[0]==1,a[n]==(3n+5)a[n-1]},a,{n,20}] (* Harvey P. Dale, Oct 19 2013 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(8/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(8/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((3*n+5)(!^3))/5(!^3).
E.g.f.: 1/(1-3*x)^(8/3).
a(n) = 3^n*(n+5/3)!/(5/3)!. - Paul Barry, Sep 04 2005
a(n) = (3*n+5)*a(n-1). - R. J. Mathar, Nov 13 2012
Sum_{n>=0} 1/a(n) = 1 + 3*(9*e)^(1/3)*(Gamma(8/3) - Gamma(8/3, 1/3)). - Amiram Eldar, Dec 23 2022

cp's OEIS Frontend

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