A051231 -id:A051231 - cp's OEIS frontend

A051232 9-factorial numbers.

1, 9, 162, 4374, 157464, 7085880, 382637520, 24106163760, 1735643790720, 140587147048320, 12652843234348800, 1252631480200531200, 135284199861657369600, 15828251383813912243200, 1994359674360552942643200, 269238556038674647256832000

Offset: 0

Wolfdieter Lang

For n >= 1, a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_9)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001
a(n) = 9*A035023(n) = Product_{k=1..n} 9*k, n >= 1; a(0) := 1.
Pi^n/a(n) is the volume of a 2n-dimensional ball with radius 1/3. - Peter Luschny, Jul 24 2012

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

Cf. A047058, A051188, A051189. a(n) = A051231(n-1, 0), A053116 (first column of triangle).

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

with(combstruct):A:=[N,{N=Cycle(Union(Z$9))},labeled]: seq(count(A,size=n+1)/9, n=0..14); # Zerinvary Lajos, Dec 05 2007

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 8, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)

a(n) = n!*9^n =: (9*n)(!^9).
E.g.f.: 1/(1-9*x).
G.f.: 1/(1 - 9*x/(1 - 9*x/(1 - 18*x/(1 - 18*x/(1 - 27*x/(1 - 27*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, Aug 09 2017
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/9).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/9). (End)

A048176 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -10, 1, 200, -30, 1, -6000, 1100, -60, 1, 240000, -50000, 3500, -100, 1, -12000000, 2740000, -225000, 8500, -150, 1, 720000000, -176400000, 16240000, -735000, 17500, -210, 1, -50400000000, 13068000000, -1313200000, 67690000, -1960000, 32200, -280, 1, 4032000000000, -1095840000000

Offset: 1

Wolfdieter Lang

a(n,m)= R_n^m(a=0,b=10) in the notation of the given reference.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n) = product(x-10*j,j=0..n-1), n >= 1, E(0,x) := 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
Also the Bell transform of the sequence (-1)^n*A051262(n) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			{1}; {-10,1}; {200,-30,1}; {-6000,1100,-60,1}; ... E(3,x) = 200*x-30*x^2+x^3.

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

First (m=1) (unsigned) column sequence is: A051262(n-1). Row sums (signed triangle): A049212(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A045757(n). b=8: A051187, b=9: A051231.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (-1)^n*n!*10^n, 9); # Peter Luschny, Jan 28 2016

rows = 9;
t = Table[(-1)^n*n!*10^n, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

a(n, m) = a(n-1, m-1) - 10*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n

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A051232 9-factorial numbers.

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