A034689 -id:A034689 - cp's OEIS frontend

A053101 a(n) = ((6n+8)(!^6))/8(!^6), related to A034689 (((6n+2)(!^6))/2 sextic, or 6-factorials).

1, 14, 280, 7280, 232960, 8852480, 389509120, 19475456000, 1090625536000, 67618783232000, 4598077259776000, 340257717223424000, 27220617377873920000, 2340973094497157120000, 215369524693738455040000

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..344

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, this sequence, A053102, A053103 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(7/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(7/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(7/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((6*n+8)(!^6))/8(!^6)= A034689(n+2)/8.

E.g.f.: 1/(1-6*x)^(7/3).

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

Original entry on oeis.org

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

A008542 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+1).

Original entry on oeis.org

1, 1, 7, 91, 1729, 43225, 1339975, 49579075, 2131900225, 104463111025, 5745471106375, 350473737488875, 23481740411754625, 1714167050058087625, 135419196954588922375, 11510631741140058401875, 1047467488443745314570625, 101604346379043295513350625

Offset: 0

Joe Keane (jgk(AT)jgk.org)

nonn

a(n), n>=1, enumerates increasing heptic (7-ary) trees with n vertices. - Wolfdieter Lang, Sep 14 2007; see a D. Callan comment on A007559 (number of increasing quarterny trees).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for sequences related to factorial numbers.

Cf. A085158, A034689, A034723, A034724, A034787, A034788, A004993, A047058, A047657, A051151.

Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A045754 (and A084947, A114799), A045755.

List([0..20], n-> Product([0..n-1], k-> (6*k+1) )); # G. C. Greubel, Aug 17 2019

[1] cat [(&*[(6*k+1): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 17 2019

a := n -> mul(6*k+1, k=0..n-1);
G(x):=(1-6*x)^(-1/6): 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..15); # Zerinvary Lajos, Apr 03 2009

Table[Product[(6*k+1), {k,0,n-1}], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008, modified by G. C. Greubel, Aug 17 2019 *)
FoldList[Times, 1, 6Range[0, 20] + 1] (* Vincenzo Librandi, Jun 10 2013 *)
Table[6^n*Pochhammer[1/6, n], {n,0,20}] (* G. C. Greubel, Aug 17 2019 *)

a(n)=prod(k=1,n-1,6*k+1) \\ Charles R Greathouse IV, Jul 19 2011

[product((6*k+1) for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 17 2019

E.g.f.: (1-6*x)^(-1/6).
a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(1/6)^-1*n^(-1/3)*6^n*e^-n*n^n*{1 + 1/72*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = Sum_{k=0..n} (-6)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005
G.f.: 1+x/(1-7x/(1-6x/(1-13x/(1-12x/(1-19x/(1-18x/(1-25x/(1-24x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-5)^n*Sum_{k=0..n} (6/5)^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*(6*k+1)/(1 - x*(6*k+6)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = A085158(6*n-5). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-6*n+5)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/6^5)^(1/6)*(Gamma(1/6) - Gamma(1/6, 1/6)). - Amiram Eldar, Dec 18 2022

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

Original entry on oeis.org

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512

Offset: 0

Dale Gerdemann, Apr 12 2015

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:
  a\b 1.......2.......3.......4.......5.......6
  -1  A144431
  0   A007318 A038208 A038221
  1   A008292 A256890 A257180 A257606 A257607
  2   A060187 A257609 A257611 A257613 A257615
  3   A142458 A257610 A257620 A257622 A257624 A257626
  4   A142459 A257612 A257621
  5   A142460 A257614 A257623
  6   A142461 A257616 A257625
  7   A142462 A257617 A257627
  8   A167884 A257618
  9   A257608 A257619
The row sums of these, and similarly constructed number triangles, are shown in the following table:
  a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
  0   A000079 A000302 A000400
  1   A000142 A001715 A001725 A049388 A049198
  2   A000165 A002866 A002866 A051580 A051582
  3   A008544 A051578 A037559 A051605 A051607 A051609
  4   A001813 A047053 A000407 A034177 A051618 A051620 A051622
  5   A047055 A008546 A008548 A034300 A034325 A051688 A051690
  6   A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
  7   A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
  8   A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
  9   A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
  10                                  A051262 A035265 A035273         A035277
  11                                  A254322
  12                                          A145448
The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).
  a\b  0           1           2          3
  -2   A130595/1
  -1
  0
  1    A110555/-1  A120434/-1  A144697/1  A144699/2
With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).
If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

			Array, t(n, k), begins as:
   1,    2,      4,        8,        16,         32,          64, ...;
   2,   12,     52,      196,       684,       2276,        7340, ...;
   4,   52,    416,     2644,     14680,      74652,      357328, ...;
   8,  196,   2644,    26440,    220280,    1623964,    10978444, ...;
  16,  684,  14680,   220280,   2643360,   27227908,   251195000, ...;
  32, 2276,  74652,  1623964,  27227908,  381190712,  4677894984, ...;
  64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
    1;
    2,     2;
    4,    12,      4;
    8,    52,     52,       8;
   16,   196,    416,     196,      16;
   32,   684,   2644,    2644,     684,      32;
   64,  2276,  14680,   26440,   14680,    2276,     64;
  128,  7340,  74652,  220280,  220280,   74652,   7340,   128;
  256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172,   256;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.

Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.

Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.

A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022

Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)

t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015

def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (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

A034723 a(n) is the n-th sextic factorial number divided by 3.

Original entry on oeis.org

1, 9, 135, 2835, 76545, 2525985, 98513415, 4433103675, 226088287425, 12887032383225, 811883040143175, 56019929769879075, 4201494732740930625, 340321073352015380625, 29607933381625338114375, 2753537804491156444636875, 272600242644624488019050625

Offset: 1

Wolfdieter Lang

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

Cf. A000108, A001147, A008542, A034689.

F:=Factorial;; List([1..20], n-> 3^(n-1)*F(2*n)/(2^n*F(n))); # G. C. Greubel, Nov 11 2019

F:=Factorial; [3^(n-1)*F(2*n)/(2^n*F(n)): n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq(3^(n-1)*(2*n)!/(2^n*n!), n=1..20); # G. C. Greubel, Nov 11 2019

Table[3^(n-1)*(2*n)!/(2^n*n!), {n,20}] (* G. C. Greubel, Nov 11 2019 *)

a(n) = prod(j=1, n, 6*j-3)/3; \\ Michel Marcus, Mar 13 2019

f=factorial; [3^(n-1)*f(2*n)/(2^n*f(n)) for n in (1..20)] # G. C. Greubel, Nov 11 2019

3*a(n) = (6*n-3)(!^6) = Product_{j=1..n} (6*j-3) = 3^n*A001147(n) = 3^n*(2*n)!/(2^n*n!).
E.g.f.: (-1 + (1-6*x)^(-1/2))/3.
a(n) = 2*(3/2)^(n-1)*(n+1)!*C(n), where C(n) = A000108(n). - G. C. Greubel, Nov 11 2019
D-finite with recurrence: a(n) + 3*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = e^(1/6)*sqrt(3*Pi/2)*erf(1/sqrt(6)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = e^(-1/6)*sqrt(3*Pi/2)*erfi(1/sqrt(6)), where erfi is the imaginary error function. (End)

A034788 a(n) is the n-th sextic factorial number divided by 6.

Original entry on oeis.org

1, 12, 216, 5184, 155520, 5598720, 235146240, 11287019520, 609499054080, 36569943244800, 2413616254156800, 173780370299289600, 13554868883344588800, 1138608986200945459200, 102474808758085091328000, 9837581640776168767488000, 1003433327359169214283776000

Offset: 1

Wolfdieter Lang

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

Cf. A008542, A034689, A034723, A034724, A034787.

List([1..20], n-> 6^(n-1)*Factorial(n) ); # G. C. Greubel, Nov 11 2019

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

seq(6^(n-1)*n!, n=1..20); # G. C. Greubel, Nov 11 2019

Table[6^(n-1)*n!,{n,20}] (* Harvey P. Dale, Dec 22 2013 *)

vector(20, n, 6^(n-1)*n!) \\ G. C. Greubel, Nov 11 2019

[6^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Nov 11 2019

6*a(n) = (6*n)(!^6) = Product_{j=1..n} 6*j = 6^n*n!.
E.g.f.: (-1 + 1/(1-6*x))/6.
D-finite with recurrence: a(n) - 6*n*a(n-1) = 0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 6*(exp(1/6)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(1-exp(-1/6)). (End)

A004117 Numerators of expansion of (1-x)^(-1/3).

Original entry on oeis.org

1, 1, 2, 14, 35, 91, 728, 1976, 5434, 135850, 380380, 1071980, 9111830, 25933670, 74096200, 637227320, 1832028545, 5280552865, 137294374490, 397431084050, 1152550143745, 10043651252635, 29217894553120, 85112997176480

Offset: 0

N. J. A. Sloane

nonn

For n >= 1, a(n) is also the numerator of beta(n+1/3,2/3)*sqrt(27)/(2*Pi). - Groux Roland, May 17 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 93.

Cf. A004128.

Cf. A004987, A007559, A047657, A054861, A034689, A053101, A072888.

Table[Numerator[Binomial[-1/3,n] (-1)^n],{n,0,40}] (* Vincenzo Librandi, Jun 13 2012 *)

a(n)=prod(k=1,n,3*k-2)/n!*3^sum(k=1,n,valuation(k,3))

(1/n!) * 3^A054861(n) * Product_{k=0..n-1} (3k+1). - Ralf Stephan, Mar 13 2004

Numerators in (1-3t)^(-1/3) = 1 + t + 2*t^2 + (14/3)*t^3 + (35/3)*t^4 + (91/3)*t^5 + (728/9)*t^6 + (1976/9)*t^7 + (5434/9)*t^8 + ... = 1 + t + 4*t^2/2! + 28*t^3/3! + 280*t^4/4! + 3640*t^5/5! + 58240*t^6/6! + ... = e.g.f. for triple factorials A007559 (cf. A094638). - Tom Copeland, Dec 04 2013

Typo in formula fixed by Pontus von Brömssen, Nov 25 2008

A053103 a(n) = ((6n+10)(!^6))/10(!^6), related to A034724 (((6n+4)(!^6))/4 sextic, or 6-factorials).

Original entry on oeis.org

1, 16, 352, 9856, 335104, 13404160, 616591360, 32062750720, 1859639541760, 119016930672640, 8331185147084800, 633170071178444800, 51919945836632473600, 4568955233623657676800, 429481791960623821619200

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..343

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, A053101, A053102, this sequence (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(8/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 15, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(16/6), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(8/3))) \\ G. C. Greubel, Aug 16 2018

a(n) = ((6*n+10)(!^6))/10(!^6) = A034724(n+2)/10.

E.g.f.: 1/(1-6*x)^(8/3).

cp's OEIS Frontend

A053101 a(n) = ((6*n+8)(!^6))/8(!^6), related to A034689 (((6*n+2)(!^6))/2 sextic, or 6-factorials).

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

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A008542 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+1).

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

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

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

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A053101 a(n) = ((6n+8)(!^6))/8(!^6), related to A034689 (((6n+2)(!^6))/2 sextic, or 6-factorials).

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

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

A053103 a(n) = ((6n+10)(!^6))/10(!^6), related to A034724 (((6n+4)(!^6))/4 sextic, or 6-factorials).