A008543 -id:A008543 - cp's OEIS frontend

A112942 INVERT transform (with offset) of sextuple factorials (A008543), where g.f. satisfies: A(x) = 1 + x[d/dx xA(x)^6]/A(x)^6.

1, 1, 6, 66, 1086, 24186, 684006, 23506626, 951191646, 44281107066, 2330310876486, 136747268000706, 8851092668419326, 626304664252772346, 48092138192079689766, 3982448437177141451586, 353746119265020213643806

Offset: 0

Paul D. Hanna, Oct 09 2005

nonn

Generally, if g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^p]/A(x)^p, then a(n) ~ (n-1)! * p^(n-1) / (Gamma((p-1)/p) * n^(1/p)). - Vaclav Kotesovec, Feb 22 2014

			A(x) = 1 + x + 6*x^2 + 66*x^3 + 1086*x^4 + 24186*x^5 +...
1/A(x) = 1 - x - 5*x^2 - 55*x^3 - 935*x^4 -... -A008543(n)*x^(n+1)-...

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

Cf. A008543, A112943 (log derivative); A112934, A112935, A112936, A112937, A112938, A112939, A112940, A112941.

CoefficientList[Series[1/(1 + 1/6*ExpIntegralE[5/6,-1/(6*x)]/E^(1/(6*x))), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 22 2014 *)

{a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x+6*x^2*deriv(F)/F); return(polcoeff(F,n,x))}

G.f. satisfies: A(x) = 1+x + 6*x^2*[d/dx A(x)]/A(x) (log derivative).
G.f.: A(x) = 1+x+6*x^2/(1-11*x-6*2*5*x^2/(1-23*x-6*3*11*x^2/(1-35*x -6*4*17*x^2/(1-47*x- ... -6*n*(6*n-7)*x^2/(1-(12*n-1)*x - ...)))) (continued fraction).
G.f.: A(x) = 1/(1-1*x/(1-5*x/(1-6*x/(1-11*x/(1-12*x/(1-17*x/(1-18*x/(1 -...)))))))) (continued fraction).
G.f.: G(0) where G(k) = 1 - x*(6*k-1)/( 1 - 6*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 24 2013
a(n) ~ (n-1)! * 6^(n-1) / (Gamma(5/6) * n^(1/6)). - Vaclav Kotesovec, Feb 22 2014

A112943 Logarithmic derivative of A112942 such that a(n)=(1/6)*A112942(n+1) for n>0, where A112942 equals the INVERT transform (with offset) of sextuple factorials A008543.

Original entry on oeis.org

1, 11, 181, 4031, 114001, 3917771, 158531941, 7380184511, 388385146081, 22791211333451, 1475182111403221, 104384110708795391, 8015356365346614961, 663741406196190241931, 58957686544170035607301

Offset: 1

Paul D. Hanna, Oct 09 2005

nonn

			log(1+x + 6*x*[x + 11*x^2 + 181*x^3 + 4031*x^4 + 114001*x^5 +...])
= x + 11/2*x^2 + 181/3*x^3 + 4031/4*x^4 + 114001/5*x^5 + ...

Cf. A008543, A112942; A112934, A112935, A112936, A112937, A112938, A112939, A112940, A112941.

{a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x+6*x^2*deriv(F)/F); return(n*polcoeff(log(F),n,x))}

G.f.: log(1+x + 6*x*[Sum_{n>=1} a(n)]) = Sum_{n>=1} a(n)/n*x^n.

A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 16, 27, 40, 55, 72, 91, 224, 405, 640, 935, 1296, 1729, 4480, 8505, 14080, 21505, 31104, 43225, 116480, 229635, 394240, 623645, 933120, 1339975, 3727360, 7577955, 13404160, 21827575, 33592320, 49579075, 141639680

Offset: 0

Hugo Pfoertner, Jun 21 2003

nonn

The term "Sextuple factorial numbers" is also used for the sequences A008542, A008543, A011781, A047058, A047657, A049308, which have a different definition. The definition given here is the one commonly used.

			a(14) = 224 because 14*a(14-6) = 14*a(8) = 14*16 = 224.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. n!:A000142, n!!:A006882, n!!!:A007661, n!!!!:A007662, n!!!!!:A085157, 6-factorial primes: n!!!!!!+1:A085150, n!!!!!!-1:A051592.

Cf. A288093.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-6);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 21 2019

b:=func< n | n le 6 select n else n*Self(n-6) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 21 2019

a:= n-> `if`(n<1, 1, n*a(n-6)); seq(a(n), n=0..40); # G. C. Greubel, Aug 21 2019

Table[Times@@Range[n,1,-6],{n,0,40}] (* Harvey P. Dale, Aug 10 2019 *)

a(n)=if(n<1, 1, n*a(n-6));
vector(40, n, n--; a(n) ) \\ G. C. Greubel, Aug 21 2019

def a(n):
    if (n<1): return 1
    else: return n*a(n-6)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 21 2019

a(n)=1 for n < 1, otherwise a(n) = n*a(n-6).

Sum_{n>=0} 1/a(n) = A288093. - Amiram Eldar, Nov 10 2020

A047058 a(n) = 6^n * n!.

Original entry on oeis.org

1, 6, 72, 1296, 31104, 933120, 33592320, 1410877440, 67722117120, 3656994324480, 219419659468800, 14481697524940800, 1042682221795737600, 81329213300067532800, 6831653917205672755200, 614848852548510547968000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

For n >= 1, a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_6)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001
a(n) is the number of ways 3 members of each of n different teams can be arranged in a row so that members of the same team are together. - Geoffrey Critzer, Mar 30 2009
From Jianing Song, Mar 29 2021: (Start)
Number of n X n monomial matrices with entries 0, +/-1, +/-w, +/-w^2, where w = (-1 + sqrt(3)*i)/2 is a primitive 3rd root of unity.
a(n) is the order of the group U_n(Z[w]) = {A in M_n(Z[w]): A*A^H = I_n}, the group of n X n unitary matrices over the Eisenstein integers. Here A^H is the conjugate transpose of A. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 528.

Cf. A000142, A000165, A008542, A008543, A047053, A051151, A053103, A092515, A092727.

[6^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 05 2011

seq( 6^n*n!, n=0..20); # G. C. Greubel, Jun 08 2020

Table[6^n n!,{n,0,20}] (* Harvey P. Dale, Mar 30 2018 *)

a(n) = A051151(n+1, 0).
E.g.f.: 1/(1 - 6*x).
G.f.: 1/(1 -6*x/(1 - 6*x/(1 - 12*x/(1 - 12*x/(1 - 18*x/(1 - 18*x/(1 - 24*x/(1 - 24*x/(1 - 30*x/(1 - 30*x/(1 -... (continued fraction). - Philippe Deléham, Jan 08 2012
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/6) (A092515).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/6) (A092727). (End)

Name changed by Arkadiusz Wesolowski, Oct 04 2011

A049209 a(n) = -Product_{k=0..n} (7*k-1); sept-factorial numbers.

Original entry on oeis.org

1, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, 155008339200, 9610517030400, 663125675097600, 50397551307417600, 4182996758515660800, 376469708266409472000, 36517561701841718784000, 3797826416991538753536000, 421558732286060801642496000

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..330
Index entries for sequences related to factorial numbers.

Cf. A034833, A045754, A048994, A051188, A084947, A144739, A144827, A147585.
Row sums of triangle A051186 (scaled Stirling1 triangle).
Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), this sequence (m=7), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

[ -&*[ (7*k-1): k in [0..n-1] ]: n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008

CoefficientList[Series[(1-7*x)^(-6/7),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
With[{m=7}, Table[m^n*Pochhammer[(m-1)/m, n], {n, 0, 30}]] (* G. C. Greubel, Feb 16 2022 *)

m=7; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

a(n) = 6*A034833(n) = (7*n-1)*(!^7), n >= 1, a(0) := 1.
a(n) = Product_{k=1..n} (7*k - 1). a(0) = 1; a(n) = (7*n - 1)*a(n-1) for n > 0. - Klaus Brockhaus, Nov 10 2008
G.f.: 1/(1-6*x/(1-7*x/(1-13*x/(1-14*x/(1-20*x/(1-21*x/(1-27*x/(1-28*x/(1-...(continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k=0..n} 7^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
a(n) = 7^n * Gamma(n+6/7) / Gamma(6/7). - Vaclav Kotesovec, Jan 28 2015
E.g.f.: (1-7*x)^(-6/7). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (14*k+6)*x - 7*(k+1)*(7*k+6)*x^2/G(k+1); (continued fraction).
which starts as 1/(1-6*x-42*x^2/(1-20*x-182*x^2/(1-34*x-420*x^2/(1-48*x-756*x^2/(1-62*x-1190*x^2/(1-... )))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - (7*k+6)*x/(1 - (7*k+7)*x/Q(k+1) ); (continued fraction). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/7)^(1/7)*(Gamma(6/7) - Gamma(6/7, 1/7)). - Amiram Eldar, Dec 19 2022

A049211 a(n) = Product_{k=1..n} (9*k - 1); 9-factorial numbers.

Original entry on oeis.org

1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 101636305971200, 9045631231436800, 886471860680806400, 94852489092846284800, 11002888734770169036800, 1375361091846271129600000, 184298386307400331366400000, 26354669241958247385395200000

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..325

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), this sequence (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

Cf. A035022, A048994.

m:=9; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022

CoefficientList[Series[(1-9*x)^(-8/9),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

a(n) = prod(k=1, n, 9*k-1); \\ Michel Marcus, Jan 08 2015

m=9; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

a(n) = 8*A035022(n) = (9*n-1)(!^9), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 9^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
a(n) = 9^n * Gamma(n+8/9) / Gamma(8/9). - Vaclav Kotesovec, Jan 28 2015
E.g.f: (1-9*x)^(-8/9). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 09 2020: (Start)
G.f.: 1/(1-8*x-72*x^2/(1-26*x-306*x^2/(1-44*x-702*x^2/(1-62*x-1260*x^2/(1-80*x-1980*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/(1-8*x/(1-9*x/(1-17*x/(1-18*x/(1-26*x/(1-27*x/(1-35*x/(1-36*x/(1-44*x/(1-45*x/(1-...))))))))))) (Stieltjes continued fraction). (End)
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (18*k+8)*x - 9*(k+1)*(9*k+8)*x^2/G(k+1) (continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(9*k+8)/(1 - x*(9*k+9)/Q(k+1) ) (continued fraction). (End)
G.f.: hypergeometric2F0([1, 8/9], [--], 9*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). - Amiram Eldar, Dec 21 2022

a(9) (originally given incorrectly as 1011636305971200) corrected by Peter Bala, Feb 20 2015
a(15)-a(16) from Vincenzo Librandi, Feb 20 2015
a(16) corrected and incorrect MAGMA program removed by Georg Fischer, May 10 2021

A049210 a(n) = -Product_{k=0..n} (8*k-1); octo-factorial numbers.

Original entry on oeis.org

1, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 33760034483175, 2667042724170825, 232032717002861775, 22043108115271868625, 2270440135873002468375, 252018855081903273989625, 29990243754746489604765375, 3808760956852804179805202625

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..333
Index entries for sequences related to factorial numbers.

Cf. A034975, A048994, A051187.

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), this sequence (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

m:=8; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..30]]; // G. C. Greubel, Feb 16 2022

FoldList[Times,1,8*Range[20]-1] (* Harvey P. Dale, Aug 03 2014 *)
CoefficientList[Series[(1-8*x)^(-7/8),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

a(n) = -prod(k=0, n, 8*k-1); \\ Michel Marcus, Jan 08 2015

m=8; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

a(n) = 7*A034975(n) = (8*n-1)(!^8), n >= 1, a(0) = 1.
G.f.: 1/(1-7*x/(1-8*x/(1-15*x/(1-16*x/(1-23*x/(1-24*x/(1-31*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (-1)^n*Sum_{k=0..n} 8^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 - 1/Q(0) )/x where Q(k) = 1 - x*(8*k-1)/(1 - x*(8*k+8)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = 8^n*Gamma(n+7/8)/Gamma(7/8). - R. J. Mathar, Mar 20 2013
E.g.f: (1-8*x)^(-7/8). - Vaclav Kotesovec, Jan 28 2015
G.f.: 1/(1-7*x-56*x^2/(1-23*x-240*x^2/(1-39*x-552*x^2/(1-55*x-992*x^2/(1-71*x-1560*x^2/(1-... )))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Dec 09 2020
G.f.: 1/G(0) where G(k) = 1 - (16*k+7)*x - 8*(k+1)*(8*k+7)*x^2/G(k+1); (continued fraction). - Nikolaos Pantelidis, Dec 19 2020
Sum_{n>=0} 1/a(n) = 1 + (e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). - Amiram Eldar, Dec 20 2022

A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0).

Original entry on oeis.org

1, 5, 1, 55, 15, 1, 935, 295, 30, 1, 21505, 7425, 925, 50, 1, 623645, 229405, 32400, 2225, 75, 1, 21827575, 8423415, 1298605, 103600, 4550, 105, 1, 894930575, 358764175, 59069010, 5235405, 271950, 8330, 140, 1, 42061737025, 17398082625, 3016869625, 289426830, 16929255, 621810, 14070, 180, 1

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A049224; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n, m) = S2p(-5; n,m), a member of a sequence of triangles including S2p(-1; n,m) = A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) = A008277(n,m) (Stirling 2nd kind). T(n, 1) = A008543(n-1).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			Triangle begins as:
          1;
          5,         1;
         55,        15,        1;
        935,       295,       30,       1;
      21505,      7425,      925,      50,      1;
     623645,    229405,    32400,    2225,     75,     1;
   21827575,   8423415,  1298605,  103600,   4550,   105,    1;
  894930575, 358764175, 59069010, 5235405, 271950,  8330,  140,   1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Peter Luschny, The Bell transform
Index entries for sequences related to Bessel functions or polynomials

Cf. A008277, A008543, A049224, A264428.
Cf. A028844 (row sums).
Triangles with the recurrence T(n,k) = (m*(n-1)-k)*T(n-1,k) + T(n-1,k-1): A010054 (m=1), A001497 (m=2), A004747 (m=3), A000369 (m=4), A011801 (m=5), this sequence (m=6).

function T(n,k) // T = A013988
  if k eq 0 then return 0;
  elif k eq n then return 1;
  else return (6*(n-1)-k)*T(n-1,k) + T(n-1,k-1);
  end if;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2023

(* First program *)
rows = 10;
b[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
A = Table[b[n, m], {n, 1, rows}, {m, 1, rows}] // Inverse // Abs;
A013988 = Table[A[[n, m]], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==0, 0, If[k==n, 1, (6*(n-1) -k)*T[n-1,k] +T[n-1, k-1]]];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 03 2023 *)

# uses[inverse_bell_matrix from A264428]
# Adds 1,0,0,0, ... as column 0 at the left side of the triangle.
inverse_bell_matrix(lambda n: factorial(n)*binomial(5, n), 8) # Peter Luschny, Jan 16 2016

T(n, m) = n!*A049224(n, m)/(m!*6^(n-m));

T(n+1, m) = (6*n-m)*T(n, m) + T(n, m-1), for n >= m >= 1, with T(n, m) = 0, n

E.g.f. of m-th column: ((1 - (1-6*x)^(1/6))^m)/m!.

Sum_{k=1..n} T(n, k) = A028844(n).

New name from Peter Luschny, Jan 16 2016

A049308 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+4).

Original entry on oeis.org

1, 4, 40, 640, 14080, 394240, 13404160, 536166400, 24663654400, 1282510028800, 74385581670400, 4760677226905600, 333247405883392000, 25326802847137792000, 2076797833465298944000, 182758209344946307072000, 17179271678424952864768000, 1717927167842495286476800000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..340

Cf. A008543, A008544, A011781, A048994, A073006.

[n le 2 select 4^(n-1) else 2*(3*n-1)*Self(n-1): n in [1..30]]; // G. C. Greubel, Mar 29 2022

Table[6^n*Pochhammer[2/3, n], {n,0,30}] (* G. C. Greubel, Mar 29 2022 *)

a(n) = prod(k=0, n-1, 6*k+4); \\ Michel Marcus, Mar 30 2022

[6^n*rising_factorial(2/3,n) for n in (0..30)] # G. C. Greubel, Mar 29 2022

E.g.f.: (1-6*x)^(-2/3).
a(n) = 2^n*A008544(n).
G.f.: 1/(1-4*x/(1-6*x/(1-10*x/(1-12*x/(1-16*x/(1-18*x/(1-22*x/(1-24*x/(1-28*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-2)^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 - 1/Q(0) )/x/2 where Q(k) = 1 - x*(6*k-2)/(1 - x*(6*k+6)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
D-finite with recurrence: a(n) = 2*(3*n-1)*a(n-1). - R. J. Mathar, Jan 17 2020
From G. C. Greubel, Mar 29 2022: (Start)
a(n) = 6^n * Pochhammer(n, 2/3).
G.f.: Hypergeometric2F0([1, 2/3], [], 6*x). (End)
Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*(Gamma(2/3) - Gamma(2/3, 1/6))/6^(1/3). - Amiram Eldar, Dec 18 2022
a(n) ~ sqrt(Pi) * 2^(n+1/2) * (3/e)^n * n^(n+1/6) / Gamma(2/3). - Amiram Eldar, Sep 01 2025

A157405 A partition product of Stirling_2 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 55, 1, 105, 220, 935, 1, 425, 3300, 4675, 21505, 1, 3075, 47850, 84150, 129030, 623645, 1, 15855, 415800, 2323475, 2709630, 4365515, 415800, 2323475, 2709630, 4365515, 21827575, 1, 123515, 6394080, 51934575

Offset: 0

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144268.
Same partition product with length statistic is A013988.
Diagonal a(A000217) = A008543.
Row sum is A028844.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(6*j - 1).

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cp's OEIS Frontend

A112942 INVERT transform (with offset) of sextuple factorials (A008543), where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^6]/A(x)^6.

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A112943 Logarithmic derivative of A112942 such that a(n)=(1/6)*A112942(n+1) for n>0, where A112942 equals the INVERT transform (with offset) of sextuple factorials A008543.

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A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.

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A047058 a(n) = 6^n * n!.

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A049209 a(n) = -Product_{k=0..n} (7*k-1); sept-factorial numbers.

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A049211 a(n) = Product_{k=1..n} (9*k - 1); 9-factorial numbers.

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A049210 a(n) = -Product_{k=0..n} (8*k-1); octo-factorial numbers.

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A112942 INVERT transform (with offset) of sextuple factorials (A008543), where g.f. satisfies: A(x) = 1 + x[d/dx xA(x)^6]/A(x)^6.