A075513 -id:A075513 - cp's OEIS frontend

A074932 Row sums of unsigned triangle A075513.

1, 3, 18, 170, 2200, 36232, 725200, 17095248, 463936896, 14246942336, 488428297984, 18491942300416, 766293946203136, 34498781924766720, 1676731077272217600, 87501958444207351808, 4880017252828686155776

Offset: 1

Wolfdieter Lang, Oct 02 2002

			E.g.f.: A(x) = x + 3*x^2/2! + 18*x^3/3! + 170*x^4/4! + 2200*x^5/5! +...
where exp(A(x)) = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! + 67328*x^6/6! +...+ A201595(n)*x^n/n! +...

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Victor J. W. Guo, Yiting Yang, Proof of a conjecture of Kløve on permutation codes under the Chebychev distance, arXiv:1704.01295 [cs.IT], 2017. Also in Designs, Codes and Cryptography, June 2017, Volume 83, Issue 3, pp 685-690.
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

Cf. A201595.

Rest[CoefficientList[Series[Log[x-LambertW[-x*Exp[x]]]-Log[2*x], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 04 2012 *)
a[n_] := Sum[Binomial[n-1, k]*(k+1)^(n-1), {k, 0, n-1}]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Jul 09 2013, after Paul D. Hanna *)

{a(n)=sum(k=0,n-1,binomial(n-1,k)*(k+1)^(n-1))} \\ Paul D. Hanna, Aug 02 2012

{a(n)=local(A201595=serreverse(x-x*tanh(x+x^2*O(x^n)))/x);n!*polcoeff(log(A201595), n)} \\ Paul D. Hanna, Aug 02 2012

{a(n) = my(A); if( n<0, 0, A = O(x); for(k=1, n, A = log( (1 + exp( 2*x * exp(A))) / 2 )); n! * polcoeff(A, n))}; /* Michael Somos, Apr 10 2018 */

a(n) = sum(|A075513(n, m)|, m=0..n-1) = sum(binomial(n-1, m)*(m+1)^(n-1), m=0..n-1), n>=1.
E.g.f.: log(G(x)) where G(x) = (1 + exp(2*x*G(x)))/2 is the e.g.f. of A201595. - Paul D. Hanna, Aug 02 2012
E.g.f: log(x-LambertW(-x*exp(x)))-log(2*x). - Vaclav Kotesovec, Dec 04 2012
a(n) ~ n!/(sqrt(2*Pi*(1+LambertW(exp(-1))))*n^(3/2)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Dec 04 2012
a(n) = A072034(n)/n. - Vladimir Reshetnikov, Nov 09 2016
O.g.f.: Sum_{k>=1} k^(k-1)*x^k/(1 - k*x)^k. - Ilya Gutkovskiy, Oct 09 2018

A000142 Factorial numbers: n! = 123*4...n (order of symmetric group S_n, number of permutations of n letters).

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 0

N. J. A. Sloane

The earliest publication that discusses this sequence appears to be the Sepher Yezirah [Book of Creation], circa AD 300. (See Knuth, also the Zeilberger link.) - N. J. A. Sloane, Apr 07 2014
For n >= 1, a(n) is the number of n X n (0,1) matrices with each row and column containing exactly one entry equal to 1.
This sequence is the BinomialMean transform of A000354. (See A075271 for definition.) - John W. Layman, Sep 12 2002 [This is easily verified from the Paul Barry formula for A000354, by interchanging summations and using the formula: Sum_k (-1)^k C(n-i, k) = KroneckerDelta(i,n). - David Callan, Aug 31 2003]
Number of distinct subsets of T(n-1) elements with 1 element A, 2 elements B, ..., n - 1 elements X (e.g., at n = 5, we consider the distinct subsets of ABBCCCDDDD and there are 5! = 120). - Jon Perry, Jun 12 2003
n! is the smallest number with that prime signature. E.g., 720 = 2^4 * 3^2 * 5. - Amarnath Murthy, Jul 01 2003
a(n) is the permanent of the n X n matrix M with M(i, j) = 1. - Philippe Deléham, Dec 15 2003
Given n objects of distinct sizes (e.g., areas, volumes) such that each object is sufficiently large to simultaneously contain all previous objects, then n! is the total number of essentially different arrangements using all n objects. Arbitrary levels of nesting of objects are permitted within arrangements. (This application of the sequence was inspired by considering leftover moving boxes.) If the restriction exists that each object is able or permitted to contain at most one smaller (but possibly nested) object at a time, the resulting sequence begins 1,2,5,15,52 (Bell Numbers?). Sets of nested wooden boxes or traditional nested Russian dolls come to mind here. - Rick L. Shepherd, Jan 14 2004
From Michael Somos, Mar 04 2004; edited by M. F. Hasler, Jan 02 2015: (Start)
Stirling transform of [2, 2, 6, 24, 120, ...] is A052856 = [2, 2, 4, 14, 76, ...].
Stirling transform of [1, 2, 6, 24, 120, ...] is A000670 = [1, 3, 13, 75, ...].
Stirling transform of [0, 2, 6, 24, 120, ...] is A052875 = [0, 2, 12, 74, ...].
Stirling transform of [1, 1, 2, 6, 24, 120, ...] is A000629 = [1, 2, 6, 26, ...].
Stirling transform of [0, 1, 2, 6, 24, 120, ...] is A002050 = [0, 1, 5, 25, 140, ...].
Stirling transform of (A165326*A089064)(1...) = [1, 0, 1, -1, 8, -26, 194, ...] is [1, 1, 2, 6, 24, 120, ...] (this sequence). (End)
First Eulerian transform of 1, 1, 1, 1, 1, 1... The first Eulerian transform transforms a sequence s to a sequence t by the formula t(n) = Sum_{k=0..n} e(n, k)s(k), where e(n, k) is a first-order Eulerian number [A008292]. - Ross La Haye, Feb 13 2005
Conjecturally, 1, 6, and 120 are the only numbers which are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005
n! is the n-th finite difference of consecutive n-th powers. E.g., for n = 3, [0, 1, 8, 27, 64, ...] -> [1, 7, 19, 37, ...] -> [6, 12, 18, ...] -> [6, 6, ...]. - Bryan Jacobs (bryanjj(AT)gmail.com), Mar 31 2005
a(n+1) = (n+1)! = 1, 2, 6, ... has e.g.f. 1/(1-x)^2. - Paul Barry, Apr 22 2005
Write numbers 1 to n on a circle. Then a(n) = sum of the products of all n - 2 adjacent numbers. E.g., a(5) = 1*2*3 + 2*3*4 + 3*4*5 + 4*5*1 +5*1*2 = 120. - Amarnath Murthy, Jul 10 2005
The number of chains of maximal length in the power set of {1, 2, ..., n} ordered by the subset relation. - Rick L. Shepherd, Feb 05 2006
The number of circular permutations of n letters for n >= 0 is 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, ... - Xavier Noria (fxn(AT)hashref.com), Jun 04 2006
a(n) is the number of deco polyominoes of height n (n >= 1; see definitions in the Barcucci et al. references). - Emeric Deutsch, Aug 07 2006
a(n) is the number of partition tableaux of size n. See Steingrimsson/Williams link for the definition. - David Callan, Oct 06 2006
Consider the n! permutations of the integer sequence [n] = 1, 2, ..., n. The i-th permutation consists of ncycle(i) permutation cycles. Then, if the Sum_{i=1..n!} 2^ncycle(i) runs from 1 to n!, we have Sum_{i=1..n!} 2^ncycle(i) = (n+1)!. E.g., for n = 3 we have ncycle(1) = 3, ncycle(2) = 2, ncycle(3) = 1, ncycle(4) = 2, ncycle(5) = 1, ncycle(6) = 2 and 2^3 + 2^2 + 2^1 + 2^2 + 2^1 + 2^2 = 8 + 4 + 2 + 4 + 2 + 4 = 24 = (n+1)!. - Thomas Wieder, Oct 11 2006
a(n) is the number of set partitions of {1, 2, ..., 2n - 1, 2n} into blocks of size 2 (perfect matchings) in which each block consists of one even and one odd integer. For example, a(3) = 6 counts 12-34-56, 12-36-45, 14-23-56, 14-25-36, 16-23-45, 16-25-34. - David Callan, Mar 30 2007
Consider the multiset M = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...] = [1, 2, 2, ..., n x 'n'] and form the set U (where U is a set in the strict sense) of all subsets N (where N may be a multiset again) of M. Then the number of elements |U| of U is equal to (n+1)!. E.g. for M = [1, 2, 2] we get U = [[], [2], [2, 2], [1], [1, 2], [1, 2, 2]] and |U| = 3! = 6. This observation is a more formal version of the comment given already by Rick L. Shepherd, Jan 14 2004. - Thomas Wieder, Nov 27 2007
For n >= 1, a(n) = 1, 2, 6, 24, ... are the positions corresponding to the 1's in decimal expansion of Liouville's constant (A012245). - Paul Muljadi, Apr 15 2008
Triangle A144107 has n! for row sums (given n > 0) with right border n! and left border A003319, the INVERTi transform of (1, 2, 6, 24, ...). - Gary W. Adamson, Sep 11 2008
Equals INVERT transform of A052186 and row sums of triangle A144108. - Gary W. Adamson, Sep 11 2008
From Abdullahi Umar, Oct 12 2008: (Start)
a(n) is also the number of order-decreasing full transformations (of an n-chain).
a(n-1) is also the number of nilpotent order-decreasing full transformations (of an n-chain). (End)
n! is also the number of optimal broadcast schemes in the complete graph K_{n}, equivalent to the number of binomial trees embedded in K_{n} (see Calin D. Morosan, Information Processing Letters, 100 (2006), 188-193). - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
Let S_{n} denote the n-star graph. The S_{n} structure consists of n S_{n-1} structures. This sequence gives the number of edges between the vertices of any two specified S_{n+1} structures in S_{n+2} (n >= 1). - K.V.Iyer, Mar 18 2009
Chromatic invariant of the sun graph S_{n-2}.
It appears that a(n+1) is the inverse binomial transform of A000255. - Timothy Hopper, Aug 20 2009
a(n) is also the determinant of a square matrix, An, whose coefficients are the reciprocals of beta function: a{i, j} = 1/beta(i, j), det(An) = n!. - Enrique Pérez Herrero, Sep 21 2009
The asymptotic expansions of the exponential integrals E(x, m = 1, n = 1) ~ exp(-x)/x*(1 - 1/x + 2/x^2 - 6/x^3 + 24/x^4 + ...) and E(x, m = 1, n = 2) ~ exp(-x)/x*(1 - 2/x + 6/x^2 - 24/x^3 + ...) lead to the factorial numbers. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009
Satisfies A(x)/A(x^2), A(x) = A173280. - Gary W. Adamson, Feb 14 2010
a(n) = G^n where G is the geometric mean of the first n positive integers. - Jaroslav Krizek, May 28 2010
Increasing colored 1-2 trees with choice of two colors for the rightmost branch of nonleaves. - Wenjin Woan, May 23 2011
Number of necklaces with n labeled beads of 1 color. - Robert G. Wilson v, Sep 22 2011
The sequence 1!, (2!)!, ((3!)!)!, (((4!)!)!)!, ..., ((...(n!)!)...)! (n times) grows too rapidly to have its own entry. See Hofstadter.
The e.g.f. of 1/a(n) = 1/n! is BesselI(0, 2*sqrt(x)). See Abramowitz-Stegun, p. 375, 9.3.10. - Wolfdieter Lang, Jan 09 2012
a(n) is the length of the n-th row which is the sum of n-th row in triangle A170942. - Reinhard Zumkeller, Mar 29 2012
Number of permutations of elements 1, 2, ..., n + 1 with a fixed element belonging to a cycle of length r does not depend on r and equals a(n). - Vladimir Shevelev, May 12 2012
a(n) is the number of fixed points in all permutations of 1, ..., n: in all n! permutations, 1 is first exactly (n-1)! times, 2 is second exactly (n-1)! times, etc., giving (n-1)!*n = n!. - Jon Perry, Dec 20 2012
For n >= 1, a(n-1) is the binomial transform of A000757. See Moreno-Rivera. - Luis Manuel Rivera Martínez, Dec 09 2013
Each term is divisible by its digital root (A010888). - Ivan N. Ianakiev, Apr 14 2014
For m >= 3, a(m-2) is the number hp(m) of acyclic Hamiltonian paths in a simple graph with m vertices, which is complete except for one missing edge. For m < 3, hp(m)=0. - Stanislav Sykora, Jun 17 2014
a(n) is the number of increasing forests with n nodes. - Brad R. Jones, Dec 01 2014
The factorial numbers can be calculated by means of the recurrence n! = (floor(n/2)!)^2 * sf(n) where sf(n) are the swinging factorials A056040. This leads to an efficient algorithm if sf(n) is computed via prime factorization. For an exposition of this algorithm see the link below. - Peter Luschny, Nov 05 2016
Treeshelves are ordered (plane) binary (0-1-2) increasing trees where the nodes of outdegree 1 come in 2 colors. There are n! treeshelves of size n, and classical Françon's bijection maps bijectively treeshelves into permutations. - Sergey Kirgizov, Dec 26 2016
Satisfies Benford's law [Diaconis, 1977; Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017
a(n) = Sum((d_p)^2), where d_p is the number of standard tableaux in the Ferrers board of the integer partition p and summation is over all integer partitions p of n. Example: a(3) = 6. Indeed, the  partitions of 3 are [3], [2,1], and [1,1,1], having 1, 2, and 1 standard tableaux, respectively; we have 1^2 + 2^2 + 1^2 = 6. - Emeric Deutsch, Aug 07 2017
a(n) is the n-th derivative of x^n. - Iain Fox, Nov 19 2017
a(n) is the number of maximum chains in the n-dimensional Boolean cube {0,1}^n in respect to the relation "precedes". It is defined as follows: for arbitrary vectors u, v of {0,1}^n, such that u = (u_1, u_2, ..., u_n) and v = (v_1, v_2, ..., v_n), "u precedes v" if u_i <= v_i, for i=1, 2, ..., n. - Valentin Bakoev, Nov 20 2017
a(n) is the number of shortest paths (for example, obtained by Breadth First Search) between the nodes (0,0,...,0) (i.e., the all-zeros vector) and (1,1,...,1) (i.e., the all-ones vector) in the graph H_n, corresponding to the n-dimensional Boolean cube {0,1}^n. The graph is defined as H_n = (V_n, E_n), where V_n is the set of all vectors of {0,1}^n, and E_n contains edges formed by each pair adjacent vectors. - Valentin Bakoev, Nov 20 2017
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = sigma(gcd(i,j)) for 1 <= i,j <= n. - Bernard Schott, Dec 05 2018
a(n) is also the number of inversion sequences of length n. A length n inversion sequence e_1, e_2, ..., e_n is a sequence of n integers such that 0 <= e_i < i. - Juan S. Auli, Oct 14 2019
The term "factorial" ("factorielle" in French) was coined by the French mathematician Louis François Antoine Arbogast (1759-1803) in 1800. The notation "!" was first used by the French mathematician Christian Kramp (1760-1826) in 1808. - Amiram Eldar, Apr 16 2021
Also the number of signotopes of rank 2, i.e., mappings X:{{1..n} choose 2}->{+,-} such that for any three indices a < b < c, the sequence X(a,b), X(a,c), X(b,c) changes its sign at most once (see Felsner-Weil reference). - Manfred Scheucher, Feb 09 2022
a(n) is also the number of labeled commutative semisimple rings with n elements. As an example the only commutative semisimple rings with 4 elements are F_4 and F_2 X F_2. They both have exactly 2 automorphisms, hence a(4)=24/2+24/2=24. - Paul Laubie, Mar 05 2024
a(n) is the number of extremely unlucky Stirling permutations of order n+1; i.e., the number of Stirling permutations of order n+1 that have exactly one lucky car. - Bridget Tenner, Apr 09 2024

			There are 3! = 1*2*3 = 6 ways to arrange 3 letters {a, b, c}, namely abc, acb, bac, bca, cab, cba.
Let n = 2. Consider permutations of {1, 2, 3}. Fix element 3. There are a(2) = 2 permutations in each of the following cases: (a) 3 belongs to a cycle of length 1 (permutations (1, 2, 3) and (2, 1, 3)); (b) 3 belongs to a cycle of length 2 (permutations (3, 2, 1) and (1, 3, 2)); (c) 3 belongs to a cycle of length 3 (permutations (2, 3, 1) and (3, 1, 2)). - _Vladimir Shevelev_, May 13 2012
G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 120*x^5 + 720*x^6 + 5040*x^7 + ...

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Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
R. P. Stanley, A combinatorial miscellany.
R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.
Einar Steingrimsson and Lauren K. Williams, Permutation tableaux and permutation patterns, arXiv:math/0507149 [math.CO], 2005-2006.
A. Umar, On the semigroups of order-decreasing finite full transformations, Proc. Roy. Soc. Edinburgh 120A (1992), 129-142.
G. Villemin's Almanach of Numbers, Factorielles.
Sage Weil, The First 999 Factorials. [Cached copy at the Wayback Machine]
Eric Weisstein's World of Mathematics, Factorial, Gamma Function, Multifactorial, Permutation, Permutation Pattern, Laguerre Polynomial, Diagonal Matrix, Chromatic Invariant.
R. W. Whitty, Rook polynomials on two-dimensional surfaces and graceful labellings of graphs, Discrete Math., 308 (2008), 674-683.
Wikipedia, Factorial.
Doron Zeilberger, King Solomon and Rabbi Ben Ezra's Evaluations of Pi and Patriarch Abraham's Analysis of an Algorithm.
Doron Zeilberger, King Solomon and Rabbi Ben Ezra's Evaluations of Pi and Patriarch Abraham's Analysis of an Algorithm. [Local copy]
Doron Zeilberger and Noam Zeilberger, Two Questions about the Fractional Counting of Partitions, arXiv:1810.12701 [math.CO], 2018.
Index entries for "core" sequences.
Index to divisibility sequences.
Index entries for sequences related to factorial numbers.
Index entries for sequences related to Benford's law.

Cf. A000165, A001044, A001563, A003422, A009445, A010050, A012245, A033312, A034886, A038507, A047920, A048631.
Factorial base representation: A007623.
Cf. A003319, A052186, A144107, A144108. - Gary W. Adamson, Sep 11 2008
Complement of A063992. - Reinhard Zumkeller, Oct 11 2008
Cf. A053657, A163176. - Jonathan Sondow, Jul 26 2009
Cf. A173280. - Gary W. Adamson, Feb 14 2010
Boustrophedon transforms: A230960, A230961.
Cf. A233589.
Cf. A245334.
A row of the array in A249026.
Cf. A001013 (multiplicative closure).
For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529.
Cf. A000169, A075513, A152917, A258773.

Axiom
```
[factorial(n) for n in 0..10]
```

List([0..22],Factorial); # Muniru A Asiru, Dec 05 2018

a000142 :: (Enum a, Num a, Integral t) => t -> a
a000142 n = product [1 .. fromIntegral n]
a000142_list = 1 : zipWith (*) [1..] a000142_list
-- Reinhard Zumkeller, Mar 02 2014, Nov 02 2011, Apr 21 2011

print([factorial(big(n)) for n in 0:28]) # Paul Muljadi, May 01 2024

a:= func< n | Factorial(n) >; [ a(n) : n in [0..10]];

A000142 := n -> n!; seq(n!,n=0..20);
spec := [ S, {S=Sequence(Z) }, labeled ]; seq(combstruct[count](spec,size=n), n=0..20);
# Maple program for computing cycle indices of symmetric groups
M:=6: f:=array(0..M): f[0]:=1: print(`n= `,0); print(f[0]); f[1]:=x[1]: print(`n= `, 1); print(f[1]); for n from 2 to M do f[n]:=expand((1/n)*add( x[l]*f[n-l],l=1..n)); print(`n= `, n); print(f[n]); od:
with(combstruct):ZL0:=[S,{S=Set(Cycle(Z,card>0))},labeled]: seq(count(ZL0,size=n),n=0..20); # Zerinvary Lajos, Sep 26 2007

Table[Factorial[n], {n, 0, 20}] (* Stefan Steinerberger, Mar 30 2006 *)
FoldList[#1 #2 &, 1, Range@ 20] (* Robert G. Wilson v, May 07 2011 *)
Range[20]! (* Harvey P. Dale, Nov 19 2011 *)
RecurrenceTable[{a[n] == n*a[n - 1], a[0] == 1}, a, {n, 0, 22}] (* Ray Chandler, Jul 30 2015 *)

a(n)=prod(i=1, n, i) \\ Felix Fröhlich, Aug 17 2014

{a(n) = if(n<0, 0, n!)}; /* Michael Somos, Mar 04 2004 */

for i in range(1, 1000):
    y = i
    for j in range(1, i):
       y *= i - j
    print(y, "\n")

import math
for i in range(1, 1000):
    math.factorial(i)
    print("")
# Ruskin Harding, Feb 22 2013

[factorial(n) for n in (1..22)] # Giuseppe Coppoletta, Dec 05 2014

(1: BigInt).to(24: BigInt).scanLeft(1: BigInt)( * ) // Alonso del Arte, Mar 02 2019

Exp(x) = Sum_{m >= 0} x^m/m!. - Mohammad K. Azarian, Dec 28 2010
Sum_{i=0..n} (-1)^i * i^n * binomial(n, i) = (-1)^n * n!. - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
Sum_{i=0..n} (-1)^i * (n-i)^n * binomial(n, i) = n!. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 10 2007
The sequence trivially satisfies the recurrence a(n+1) = Sum_{k=0..n} binomial(n,k) * a(k)*a(n-k). - Robert FERREOL, Dec 05 2009
D-finite with recurrence: a(n) = n*a(n-1), n >= 1. n! ~ sqrt(2*Pi) * n^(n+1/2) / e^n (Stirling's approximation).
a(0) = 1, a(n) = subs(x = 1, (d^n/dx^n)(1/(2-x))), n = 1, 2, ... - Karol A. Penson, Nov 12 2001
E.g.f.: 1/(1-x). - Michael Somos, Mar 04 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*A000522(k)*binomial(n, k) = Sum_{k=0..n} (-1)^(n-k)*(x+k)^n*binomial(n, k). - Philippe Deléham, Jul 08 2004
Binomial transform of A000166. - Ross La Haye, Sep 21 2004
a(n) = Sum_{i=1..n} ((-1)^(i-1) * sum of 1..n taken n - i at a time) - e.g., 4! = (1*2*3 + 1*2*4 + 1*3*4 + 2*3*4) - (1*2 + 1*3 + 1*4 + 2*3 + 2*4 + 3*4) + (1 + 2 + 3 + 4) - 1 = (6 + 8 + 12 + 24) - (2 + 3 + 4 + 6 + 8 + 12) + 10 - 1 = 50 - 35 + 10 - 1 = 24. - Jon Perry, Nov 14 2005
a(n) = (n-1)*(a(n-1) + a(n-2)), n >= 2. - Matthew J. White, Feb 21 2006
1 / a(n) = determinant of matrix whose (i,j) entry is (i+j)!/(i!(j+1)!) for n > 0. This is a matrix with Catalan numbers on the diagonal. - Alexander Adamchuk, Jul 04 2006
Hankel transform of A074664. - Philippe Deléham, Jun 21 2007
For n >= 2, a(n-2) = (-1)^n*Sum_{j=0..n-1} (j+1)*Stirling1(n,j+1). - Milan Janjic, Dec 14 2008
From Paul Barry, Jan 15 2009: (Start)
G.f.: 1/(1-x-x^2/(1-3x-4x^2/(1-5x-9x^2/(1-7x-16x^2/(1-9x-25x^2... (continued fraction), hence Hankel transform is A055209.
G.f. of (n+1)! is 1/(1-2x-2x^2/(1-4x-6x^2/(1-6x-12x^2/(1-8x-20x^2... (continued fraction), hence Hankel transform is A059332. (End)
a(n) = Product_{p prime} p^(Sum_{k > 0} floor(n/p^k)) by Legendre's formula for the highest power of a prime dividing n!. - Jonathan Sondow, Jul 24 2009
a(n) = A053657(n)/A163176(n) for n > 0. - Jonathan Sondow, Jul 26 2009
It appears that a(n) = (1/0!) + (1/1!)*n + (3/2!)*n*(n-1) + (11/3!)*n*(n-1)*(n-2) + ... + (b(n)/n!)*n*(n-1)*...*2*1, where a(n) = (n+1)! and b(n) = A000255. - Timothy Hopper, Aug 12 2009
Sum_{n >= 0} 1/a(n) = e. - Jaume Oliver Lafont, Mar 03 2009
a(n) = a(n-1)^2/a(n-2) + a(n-1), n >= 2. - Jaume Oliver Lafont, Sep 21 2009
a(n) = Gamma(n+1). - Enrique Pérez Herrero, Sep 21 2009
a(n) = A173333(n,1). - Reinhard Zumkeller, Feb 19 2010
a(n) = A_{n}(1) where A_{n}(x) are the Eulerian polynomials. - Peter Luschny, Aug 03 2010
a(n) = n*(2*a(n-1) - (n-1)*a(n-2)), n > 1. - Gary Detlefs, Sep 16 2010
1/a(n) = -Sum_{k=1..n+1} (-2)^k*(n+k+2)*a(k)/(a(2*k+1)*a(n+1-k)). - Groux Roland, Dec 08 2010
From Vladimir Shevelev, Feb 21 2011: (Start)
a(n) = Product_{p prime, p <= n} p^(Sum_{i >= 1} floor(n/p^i)).
The infinitary analog of this formula is: a(n) = Product_{q terms of A050376 <= n} q^((n)_q), where (n)_q denotes the number of those numbers <= n for which q is an infinitary divisor (for the definition see comment in A037445). (End)
The terms are the denominators of the expansion of sinh(x) + cosh(x). - Arkadiusz Wesolowski, Feb 03 2012
G.f.: 1 / (1 - x / (1 - x / (1 - 2*x / (1 - 2*x / (1 - 3*x / (1 - 3*x / ... )))))). - Michael Somos, May 12 2012
G.f. 1 + x/(G(0)-x) where G(k) = 1 - (k+1)*x/(1 - x*(k+2)/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 14 2012
G.f.: W(1,1;-x)/(W(1,1;-x) - x*W(1,2;-x)), where W(a,b,x) = 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! - ... + a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! + ...; see [A. N. Khovanskii, p. 141 (10.19)]. - Sergei N. Gladkovskii, Aug 15 2012
From Sergei N. Gladkovskii, Dec 26 2012: (Start)
G.f.: A(x) = 1 + x/(G(0) - x) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1); (continued fraction).
Let B(x) be the g.f. for A051296, then A(x) = 2 - 1/B(x). (End)
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+1)/(1-x/(x - 1/(1 - (2*k+2)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: 1 + x*(1 - G(0))/(sqrt(x)-x) where G(k) = 1 - (k+1)*sqrt(x)/(1-sqrt(x)/(sqrt(x)-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 25 2013
G.f.: 1 + x/G(0) where G(k) = 1 - x*(k+2)/( 1 - x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
a(n) = det(S(i+1, j), 1 <= i, j <=n ), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: G(0), where G(k) = 1 + x*(2*k+1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
a(n) = P(n-1, floor(n/2)) * floor(n/2)! * (n - (n-2)*((n+1) mod 2)), where P(n, k) are the k-permutations of n objects, n > 0. - Wesley Ivan Hurt, Jun 07 2013
a(n) = a(n-2)*(n-1)^2 + a(n-1), n > 1. - Ivan N. Ianakiev, Jun 18 2013
a(n) = a(n-2)*(n^2-1) - a(n-1), n > 1. - Ivan N. Ianakiev, Jun 30 2013
G.f.: 1 + x/Q(0), m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
a(n) = A245334(n,n). - Reinhard Zumkeller, Aug 31 2014
a(n) = Product_{i = 1..n} A014963^floor(n/i) = Product_{i = 1..n} A003418(floor(n/i)). - Matthew Vandermast, Dec 22 2014
a(n) = round(Sum_{k>=1} log(k)^n/k^2), for n>=1, which is related to the n-th derivative of the Riemann zeta function at x=2 as follows: round((-1)^n * zeta^(n)(2)). Also see A073002. - Richard R. Forberg, Dec 30 2014
a(n) ~ Sum_{j>=0} j^n/e^j, where e = A001113. When substituting a generic variable for "e" this infinite sum is related to Eulerian polynomials. See A008292. This approximation of n! is within 0.4% at n = 2. See A255169. Accuracy, as a percentage, improves rapidly for larger n. - Richard R. Forberg, Mar 07 2015
a(n) = Product_{k=1..n} (C(n+1, 2)-C(k, 2))/(2*k-1); see Masanori Ando link. - Michel Marcus, Apr 17 2015
Sum_{n>=0} a(n)/(a(n + 1)*a(n + 2)) = Sum_{n>=0} 1/((n + 2)*(n + 1)^2*a(n)) = 2 - exp(1) - gamma + Ei(1) = 0.5996203229953..., where gamma = A001620, Ei(1) = A091725. - Ilya Gutkovskiy, Nov 01 2016
a(2^n) = 2^(2^n - 1) * 1!! * 3!! * 7!! * ... * (2^n - 1)!!. For example, 16! = 2^15*(1*3)*(1*3*5*7)*(1*3*5*7*9*11*13*15) = 20922789888000. - Peter Bala, Nov 01 2016
a(n) = sum(prod(B)), where the sum is over all subsets B of {1,2,...,n-1} and where prod(B) denotes the product of all the elements of set B. If B is a singleton set with element b, then we define prod(B)=b, and, if B is the empty set, we define prod(B) to be 1. For example, a(4)=(1*2*3)+(1*2)+(1*3)+(2*3)+(1)+(2)+(3)+1=24. - Dennis P. Walsh, Oct 23 2017
Sum_{n >= 0} 1/(a(n)*(n+2)) = 1. - Multiplying the denominator by (n+2) in Jaume Oliver Lafont's entry above creates a telescoping sum. - Fred Daniel Kline, Nov 08 2020
O.g.f.: Sum_{k >= 0} k!*x^k = Sum_{k >= 0} (k+y)^k*x^k/(1 + (k+y)*x)^(k+1) for arbitrary y. - Peter Bala, Mar 21 2022
E.g.f.: 1/(1 + LambertW(-x*exp(-x))) = 1/(1-x), see A258773. -(1/x)*substitute(z = x*exp(-x), z*(d/dz)LambertW(-z)) = 1/(1 - x). See A075513. Proof: Use the compositional inverse (x*exp(-x))^[-1] = -LambertW(-z). See A000169 or A152917, and Richard P. Stanley: Enumerative Combinatorics, vol. 2,  p. 37, eq. (5.52). - Wolfdieter Lang, Oct 17 2022
Sum_{k >= 1} 1/10^a(k) = A012245 (Liouville constant). - Bernard Schott, Dec 18 2022
From David Ulgenes, Sep 19 2023: (Start)
1/a(n) = (e/(2*Pi*n)*Integral_{x=-oo..oo} cos(x-n*arctan(x))/(1+x^2)^(n/2) dx). Proof: take the real component of Laplace's integral for 1/Gamma(x).
a(n) = Integral_{x=0..1} e^(-t)*LerchPhi(1/e, -n, t) dt. Proof: use the relationship Gamma(x+1) = Sum_{n >= 0} Integral_{t=n..n+1} e^(-t)t^x dt = Sum_{n >= 0} Integral_{t=0..1} e^(-(t+n))(t+n)^x dt and interchange the order of summation and integration.
Conjecture: a(n) = 1/(2*Pi)*Integral_{x=-oo..oo}(n+i*x+1)!/(i*x+1)-(n+i*x-1)!/(i*x-1)dx. (End)
a(n) = floor(b(n)^n / (floor(((2^b(n) + 1) / 2^n)^b(n)) mod 2^b(n))), where b(n) = (n + 1)^(n + 2) = A007778(n+1). Joint work with Mihai Prunescu. - Lorenzo Sauras Altuzarra, Oct 18 2023
a(n) = e^(Integral_{x=1..n+1} Psi(x) dx) where Psi(x) is the digamma function. - Andrea Pinos, Jan 10 2024
a(n) = Integral_{x=0..oo} e^(-x^(1/n)) dx, for n > 0. - Ridouane Oudra, Apr 20 2024
O.g.f.: N(x) = hypergeometric([1,1], [], x) = LaplaceTransform(x/(1-x))/x, satisfying x^2*N'(x) + (x-1)*N(x) + 1 = 0, with N(0) = 1. - Wolfdieter Lang, May 31 2025

A000169 Number of labeled rooted trees with n nodes: n^(n-1).

Original entry on oeis.org

1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601, 743008370688, 23298085122481, 793714773254144, 29192926025390625, 1152921504606846976, 48661191875666868481, 2185911559738696531968, 104127350297911241532841, 5242880000000000000000000

Offset: 1

N. J. A. Sloane

Also the number of connected transitive subtree acyclic digraphs on n vertices. - Robert Castelo, Jan 06 2001
For any given integer k, a(n) is also the number of functions from {1,2,...,n} to {1,2,...,n} such that the sum of the function values is k mod n. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 16 2002
The n-th term of a geometric progression with first term 1 and common ratio n: a(1) = 1 -> 1,1,1,1,... a(2) = 2 -> 1,2,... a(3) = 9 -> 1,3,9,... a(4) = 64 -> 1,4,16,64,... - Amarnath Murthy, Mar 25 2004
All rational solutions to the equation x^y = y^x, with x < y, are given by x = A000169(n+1)/A000312(n), y = A000312(n+1)/A007778(n), where n = 1, 2, 3, ... . - Nick Hobson, Nov 30 2006
a(n+1) is also the number of partial functions on n labeled objects. - Franklin T. Adams-Watters, Dec 25 2006
In other words, if A is a finite set of size n-1, then a(n) is the number of binary relations on A that are also functions. Note that a(n) = Sum_{k=0..n-1} binomial(n-1,k)*(n-1)^k = n^(n-1), where binomial(n-1,k) is the number of ways to select a domain D of size k from A and (n-1)^k is the number of functions from D to A. - Dennis P. Walsh, Apr 21 2011
This is the fourth member of a set of which the other members are the symmetric group, full transformation semigroup, and symmetric inverse semigroup. For the first three, see A000142, A000312, A002720. - Peter J. Cameron, Nov 03 2024.
More generally, consider the class of sequences of the form a(n) = (n*c(1)*...*c(i))^(n-1). This sequence has c(1)=1. A052746 has a(n) = (2*n)^(n-1), A052756 has a(n) = (3*n)^(n-1), A052764 has a(n) = (4*n)^(n-1), A052789 has a(n) = (5*n)^(n-1) for n>0. These sequences have a combinatorial structure like simple grammars. - Ctibor O. Zizka, Feb 23 2008
a(n) is equal to the logarithmic transform of the sequence b(n) = n^(n-2) starting at b(2). - Kevin Hu (10thsymphony(AT)gmail.com), Aug 23 2010
Also, number of labeled connected multigraphs of order n without cycles except one loop. See link below to have a picture showing the bijection between rooted trees and multigraphs of this kind. (Note that there are no labels in the picture, but the bijection remains true if we label the nodes.) - Washington Bomfim, Sep 04 2010
a(n) is also the number of functions f:{1,2,...,n} -> {1,2,...,n} such that f(1) = 1.
For a signed version of A000169 arising from the Vandermonde determinant of (1,1/2,...,1/n), see the Mathematica section. - Clark Kimberling, Jan 02 2012
Numerator of (1+1/(n-1))^(n-1) for n>1. - Jean-François Alcover, Jan 14 2013
Right edge of triangle A075513. - Michel Marcus, May 17 2013
a(n+1) is the number of n x n binary matrices with no more than a single one in each row. Partitioning the set of such matrices by the number k of rows with a one, we obtain a(n+1) = Sum_{k=0..n} binomial(n,k)*n^k = (n+1)^n. - Dennis P. Walsh, May 27 2014
Central terms of triangle A051129: a(n) = A051129(2*n-1,n). - Reinhard Zumkeller, Sep 14 2014
a(n) is the row sum of the n-th rows of A248120 and A055302, so it enumerates the monomials in the expansion of [x(1) + x(2) + ... + x(n)]^(n-1). - Tom Copeland, Jul 17 2015
For any given integer k, a(n) is the number of sums x_1 + ... + x_m = k (mod n) such that: x_1, ..., x_m are nonnegative integers less than n, the order of the summands does not matter, and each integer appears fewer than n times as a summand. - Carlo Sanna, Oct 04 2015
a(n) is the number of words of length n-1 over an alphabet of n letters. - Joerg Arndt, Oct 07 2015
a(n) is the number of parking functions whose largest element is n and length is n. For example, a(3) = 9 because there are nine such parking functions, namely (1,2,3), (1,3,2), (2,3,1), (2,1,3), (3,1,2), (3,2,1), (1,1,3), (1,3,1), (3,1,1). - Ran Pan, Nov 15 2015
Consider the following problem: n^2 cells are arranged in a square array. A step can be defined as going from one cell to the one directly above it, to the right of it or under it. A step above cannot be followed by a step below and vice versa. Once the last column of the square array is reached, you can only take steps down. a(n) is the number of possible paths (i.e., sequences of steps) from the cell on the bottom left to the cell on the bottom right. - Nicolas Nagel, Oct 13 2016
The rationals c(n) = a(n+1)/a(n), n >= 1, appear in the proof of G. Pólya's "elementary, but not too elementary, theorem": Sum_{n>=1} (Product_{k=1..n} a_k)^(1/n) < exp(1)*Sum_{n>=1} a_n, for n >= 1, with the sequence {a_k}{k>=1} of nonnegative terms, not all equal to 0. - _Wolfdieter Lang, Mar 16 2018
Coefficients of the generating series for the preLie operadic algebra. Cf. p. 417 of the Loday et al. paper. - Tom Copeland, Jul 08 2018
a(n)/2^(n-1) is the square of the determinant of the n X n matrix M_n with elements m(j,k) = cos(Pi*j*k/n). See Zhi-Wei Sun, Petrov link. - Hugo Pfoertner, Sep 19 2021
a(n) is the determinant of the n X n matrix P_n such that, when indexed [0, n), P(0, j) = 1, P(i <= j) = i, and P(i > j) = i-n. - C.S. Elder, Mar 11 2024

			For n=3, a(3)=9 because there are exactly 9 binary relations on A={1, 2} that are functions, namely: {}, {(1,1)}, {(1,2)}, {(2,1)}, {(2,2)}, {(1,1),(2,1)}, {(1,1),(2,2)}, {(1,2),(2,1)} and {(1,2),(2,2)}. - _Dennis P. Walsh_, Apr 21 2011
G.f. = x + 2*x^2 + 9*x^3 + 64*x^4 + 625*x^5 + 7776*x^6 + 117649*x^7 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 169.
Jonathan L. Gross and Jay Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 524.
Hannes Heikinheimo, Heikki Mannila and Jouni K. Seppnen, Finding Trees from Unordered 01 Data, in Knowledge Discovery in Databases: PKDD 2006, Lecture Notes in Computer Science, Volume 4213/2006, Springer-Verlag. - N. J. A. Sloane, Jul 09 2009
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 63.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 128.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 25, Prop. 5.3.2, and p. 37, (5.52).

N. J. A. Sloane, Table of n, a(n) for n = 1..100
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012
Washington Bomfim, Bijection between rooted forests and multigraphs without cycles except one loop in each connected component. [From _Washington Bomfim_, Sep 04 2010]
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Peter J. Cameron and Philippe Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.
Robert Castelo and Arno Siebes, A characterization of moral transitive directed acyclic graph Markov models as trees, Technical Report CS-2000-44, Faculty of Computer Science, University of Utrecht.
Robert Castelo and Arno Siebes, A characterization of moral transitive acyclic directed graph Markov models as labeled trees, Journal of Statistical Planning and Inference, Vol. 115, No. 1 (2003), pp. 235-259; alternative link.
Frédéric Chapoton, Florent Hivert and Jean-Christophe Novelli, A set-operad of formal fractions and dendriform-like sub-operads, Journal of Algebra, Vol. 465 (2016), pp. 322-355; arXiv preprint, arXiv:1307.0092 [math.CO], 2013.
Ali Chouria, Vlad-Florin Drǎgoi, Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W Function, Advances in Computational Mathematics, VOl. 5 (1996), pp. 329-359; alternative link.
Nick Hobson, Solution to puzzle 48: Exponential equation.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 67.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See p. 33.
Jean-Louis Loday and Bruno Vallette, Algebraic Operads, version 0.99, 2012.
G. Pólya, With, or Without, Motivation?, Amer. Math. Monthly, Vol. 56, No. 10 (1949), pp. 684-691. Reprinted in "A Century of Mathematics", John Ewing (ed.), Math. Assoc. of Amer., 1994, pp. 195-200 (the reference there is wrong).
Gwenaël Richomme, Characterization of infinite LSP words and endomorphisms preserving the LSP property, International Journal of Foundations of Computer Science, Vol. 30, No. 1 (2019), pp. 171-196; arXiv preprint, arXiv:1808.02680 [cs.DM], 2018.
Marko Riedel, math.stackexchange.com, Proof of an identity relating the tree function T(z) and the second order Eulerian numbers. Feb. 28, 2021.
Marko Riedel, math.stackexchange.com, Asymptotics of tree function statistics using Pusieux series
Frank Ruskey, Information on Rooted Trees.
N. J. A. Sloane, Illustration of initial terms
Zhi-Wei Sun, Fedor Petrov, A surprising identity, discussion in MathOverflow, Jan 17 2019.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.
Eric Weisstein's World of Mathematics, Graph Vertex.
Dimitri Zvonkine, An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere, arXiv:math/0403092 [math.AG], 2004.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for "core" sequences

Cf. A000055, A000081, A000142, A000272, A000312, A002720, A007778, A007830, A008785-A008791, A055860, A002061, A052746, A052756, A052764, A052789, A051129, A098686, A247363, A055302, A248120, A130293, A053506-A053509, A262974.

Other classes of endofunctions: A275549-A275558.

a000169 n = n ^ (n - 1)  -- Reinhard Zumkeller, Sep 14 2014

[n^(n-1): n in [1..20]]; // Vincenzo Librandi, Jul 17 2015

A000169 := n -> n^(n-1);
# second program:
spec := [A, {A=Prod(Z, Set(A))}, labeled]; [seq(combstruct[count](spec, size=n), n=1..20)];
# third program:
A000169 := n -> add((-1)^(n+k-1)*pochhammer(n, k)*Stirling2(n-1, k), k = 0..n-1):
seq(A000169(n), n = 1 .. 23);  # Mélika Tebni, May 07 2023

Table[n^(n - 1), {n, 1, 20}] (* Stefan Steinerberger, Apr 01 2006 *)
Range[0, 18]! CoefficientList[ Series[ -LambertW[-x], {x, 0, 18}], x] // Rest (* Robert G. Wilson v, updated by Jean-François Alcover, Oct 14 2019 *)
(* Next, a signed version A000169 from the Vandermonde determinant of (1,1/2,...,1/n) *)
f[j_] := 1/j; z = 12;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]
1/%  (* A203421 *)
Table[v[n]/v[n + 1], {n, 1, z - 1}]  (* A000169 signed *)
(* Clark Kimberling, Jan 02 2012 *)
a[n_]:=Det[Table[If[i==0,1,If[i<=j,i,i-n]],{i,0,n-1},{j,0,n-1}]]; Array[a,20] (* Stefano Spezia, Mar 12 2024 *)

n^(n-1) $ n=1..20 /* Zerinvary Lajos, Apr 01 2007 */

PARI
```
a(n) = n^(n-1)
```

def a(n): return n**(n-1)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Sep 19 2021

from sympy import Matrix
def P(n): return [[ (i-n if i > j else i) + (i == 0) for j in range(n) ] for i in range(n)]
print(*(Matrix(P(n)).det() for n in range(1, 21)), sep=', ') # C.S. Elder, Mar 12 2024

The e.g.f. T(x) = Sum_{n>=1} n^(n-1)*x^n/n! satisfies T(x) = x*exp(T(x)), so T(x) is the functional inverse (series reversion) of x*exp(-x).
Also T(x) = -LambertW(-x) where W(x) is the principal branch of Lambert's function.
T(x) is sometimes called Euler's tree function.
a(n) = A000312(n-1)*A128434(n,1)/A128433(n,1). - Reinhard Zumkeller, Mar 03 2007
E.g.f.: LambertW(x)=x*G(0); G(k) = 1 - x*((2*k+2)^(2*k))/(((2*k+1)^(2*k)) - x*((2*k+1)^(2*k))*((2*k+3)^(2*k+1))/(x*((2*k+3)^(2*k+1)) - ((2*k+2)^(2*k+1))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 30 2011
a(n) = Sum_{i=1..n} binomial(n-1,i-1)*i^(i-2)*(n-i)^(n-i). - Dmitry Kruchinin, Oct 28 2013
Limit_{n->oo} a(n)/A000312(n-1) = e. - Daniel Suteu, Jul 23 2016
From Amiram Eldar, Nov 20 2020: (Start)
Sum_{n>=1} 1/a(n) = A098686.
Sum_{n>=1} (-1)^(n+1)/a(n) = A262974. (End)
a(n) = Sum_{k=0..n-1} (-1)^(n+k-1)*Pochhammer(n, k)*Stirling2(n-1, k). - Mélika Tebni, May 07 2023
In terms of Eulerian numbers A340556(n,k) of the second order Sum_{m>=1} m^(m+n) z^m/m! = 1/(1-T(z))^(2n+1) * Sum_{k=0..n} A2(n,k) T(z)^k. - Marko Riedel, Jan 10 2024

A027472 Third convolution of the powers of 3 (A000244).

Original entry on oeis.org

1, 9, 54, 270, 1215, 5103, 20412, 78732, 295245, 1082565, 3897234, 13817466, 48361131, 167403915, 573956280, 1951451352, 6586148313, 22082967873, 73609892910, 244074908070, 805447196631, 2646469360359, 8661172452084, 28242953648100, 91789599356325, 297398301914493, 960825283108362, 3095992578904722

Offset: 3

Olivier Gérard

Third column of A027465.

With offset = 2, a(n) is the number of length n words on alphabet {u,v,w,z} such that each word contains exactly 2 u's. - Zerinvary Lajos, Dec 29 2007

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A000244, A006043, A027465, A075513, A152818.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), this sequence (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[3^(n-3)*Binomial(n-1, 2): n in [3..40]]; // G. C. Greubel, May 12 2021

nn=41; Drop[Range[0,nn]!CoefficientList[Series[Exp[x]^3 x^2/2!,{x,0,nn}],x],2] (* Geoffrey Critzer, Oct 03 2013 *)
LinearRecurrence[{9,-27,27}, {1,9,54}, 40] (* G. C. Greubel, May 12 2021 *)
Abs[Take[CoefficientList[Series[1/(1+3x^2)^3,{x,0,60}],x],{1,-1,2}]] (* Harvey P. Dale, Mar 03 2022 *)

a(n)=([0,1,0; 0,0,1; 27,-27,9]^(n-3)*[1;9;54])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[3^(n-3)*binomial(n-1,2) for n in range(3, 40)] # Zerinvary Lajos, Mar 10 2009

Numerators of sequence a[3,n] in (b^2)[i,j]) where b[i,j] = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 if j > i.
From Wolfdieter Lang: (Start)
a(n) = 3^(n-3)*binomial(n-1, 2).
G.f.: (x/(1-3*x))^3. (Third convolution of A000244, powers of 3.) (End)
a(n) = |A075513(n, 2)|/9, n >= 3.
a(n) = A152818(n-3,2)/2 = A006043(n-3)/2. - Paul Curtz, Jan 07 2009
The sequence 0, 1, 9, 54, ... has e.g.f.: (x + 3*x^2/2)*exp(3*x)/. - Paul Barry, Jul 23 2003
E.g.f.: E(0) where E(k) = 1 + 3*(2*k+3)*x/((2*k+1)^2 - 3*x*(k+2)*(2*k+1)^2/(3*x*(k+2) + 2*(k+1)^2/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
With offset=2 e.g.f.: x^2*exp(3*x)/2. - Geoffrey Critzer, Oct 03 2013
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=3} 1/a(n) = 6 - 12*log(3/2).
Sum_{n>=3} (-1)^(n+1)/a(n) = 24*log(4/3) - 6. (End)

Corrected by T. D. Noe, Nov 07 2006
Better name from Wolfdieter Lang
Terms a(23) onward added by G. C. Greubel, May 12 2021

A075497 Stirling2 triangle with scaled diagonals (powers of 2).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 28, 12, 1, 16, 120, 100, 20, 1, 32, 496, 720, 260, 30, 1, 64, 2016, 4816, 2800, 560, 42, 1, 128, 8128, 30912, 27216, 8400, 1064, 56, 1, 256, 32640, 193600, 248640, 111216, 21168, 1848, 72, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(2*z) - 1)*x/2) - 1.
Subtriangle of (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, 12, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 13 2013
Also the inverse Bell transform of the double factorial of even numbers Product_ {k=0..n-1} (2*k+2) (A000165). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
This is the exponential Riordan array [exp(2*x), (exp(2*x) - 1)/2] belonging to the derivative subgroup of the exponential Riordan group. In the notation of Corcino, this is the triangle of (2, 2)-Stirling numbers of the second kind. A factorization of the array as an infinite product is given in the example section. - Peter Bala, Feb 20 2025

			Triangle begins:
  [1];
  [2,1];
  [4,6,1]; p(3,x) = x*(4 + 6*x + x^2).
  ...;
Triangle (0, 2, 0, 4, 0, 6, 0, 8, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
  1
  0,  1
  0,  2,   1
  0,  4,   6,   1
  0,  8,  28,  12,  1
  0, 16, 120, 100, 20, 1. - _Philippe Deléham_, Feb 13 2013
From _Peter Bala_, Feb 23 2025: (Start)
The array factorizes as
/ 1               \       /1             \ /1             \ /1            \
| 2    1           |     | 2   1          ||0  1           ||0  1          |
| 4    6   1       |  =  | 4   4   1      ||0  2   1       ||0  0  1       | ...
| 8   28  12   1   |     | 8  12   6  1   ||0  4   4  1    ||0  0  2  1    |
|16  120 100  20  1|     |16  32  24  8  1||0  8  12  6  1 ||0  0  4  4  1 |
|...               |     |...             ||...            ||...           |
where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 2*x), x/(1 - 2*x)) = P^2, where P denotes Pascal's triangle. See A038207. Cf. A143494. (End)

Alois P. Heinz, Rows n = 1..141, flattened
Peter Bala, The white diamond product of power series
Peter Bala, Factorising (r,b)-Stirling arrays
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv 1507.04803 [math.CO], 2015.
Roberto B. Corcino, The (r, β)-Stirling Numbers, The Mindanao Forum, Vol. XIV, No.2, pp. 91-99, 1999.
Roberto B. Corcino and Maribeth B. Montero, The (r, β)-Stirling Numbers in the Context of 0-1 Tableau, Jour. Math. Soc. of the Philippines, ISSN 0115-6926, Vol. 32, No. 1 (2009), pp. 45-52
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
Wolfdieter Lang, First 10 rows.
Toufik Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
Emanuele Munarini, Characteristic, admittance and matching polynomials of an antiregular graph, Appl. Anal. Discrete Math 3 (1) (2009) 157-176.

Columns 1-7 are A000079, A006516, A016283, A025966, A075510-A075512.
Row sums are A004211.
Cf. A008277, A075498-A075505.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
       `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
        binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 13 2015
# Alternatively, giving the triangle in the form displayed in the Example section:
gf := exp(x*exp(z)*sinh(z)):
X := n -> series(gf, z, n+2):
Z := n -> n!*expand(simplify(coeff(X(n), z, n))):
A075497_row := n -> op(PolynomialTools:-CoefficientList(Z(n), x)):
seq(A075497_row(n), n=0..9); # Peter Luschny, Jan 14 2018

Table[(2^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)

for(n=1, 11, for(m=1, n, print1(2^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

# uses[inverse_bell_transform from A265605]
multifact_2_2 = lambda n: prod(2*k + 2 for k in (0..n-1))
inverse_bell_matrix(multifact_2_2, 9) # Peter Luschny, Dec 31 2015

a(n, m) = (2^(n-m)) * Stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*2)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 2*m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-2*k*x), m >= 1.
E.g.f. for m-th column: (((exp(2*x)-1)/2)^m)/m!, m >= 1.
The row polynomials in t are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+2*x)*d/dx. Cf. A008277. - Peter Bala, Nov 25 2011
From Peter Bala, Jan 13 2018: (Start)
n-th row polynomial R(n,x)= x o x o ... o x (n factors), where o is the deformed Hadamard product of power series defined in Bala, section 3.1.
R(n+1,x)/x = (x + 2) o (x + 2) o...o (x + 2) (n factors).
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*2^(n-k)*R(k,x).
Dobinski-type formulas: R(n,x) = exp(-x/2)*Sum_{i >= 0} (2*i)^n* (x/2)^i/i!; 1/x*R(n+1,x) = exp(-x/2)*Sum_{i >= 0} (2 + 2*i)^n* (x/2)^i/i!. (End)

A075498 Stirling2 triangle with scaled diagonals (powers of 3).

Original entry on oeis.org

1, 3, 1, 9, 9, 1, 27, 63, 18, 1, 81, 405, 225, 30, 1, 243, 2511, 2430, 585, 45, 1, 729, 15309, 24381, 9450, 1260, 63, 1, 2187, 92583, 234738, 137781, 28350, 2394, 84, 1, 6561, 557685, 2205225, 1888110, 563031, 71442, 4158, 108, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(3*z) - 1)*x/3) - 1.
Subtriangle of the triangle given by (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938, see example. - Philippe Deléham, Feb 13 2013
Also the Bell transform of A000244. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			[1]; [3,1]; [9,9,1]; ...; p(3,x) = x*(9 + 9*x + x^2).
From _Philippe Deléham_, Feb 13 2013: (Start)
Triangle (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
  1;
  0,   1;
  0,   3,   1;
  0,   9,   9,   1;
  0,  27,  63,  18,   1;
  0,  81, 405, 225,  30,   1;
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000244, A016137, A017933, A028085, A075515, A075516, A075906. Row sums are A004212.

Cf. A075497, A075499.

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 3^n, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[3^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
rows = 9;
t = Table[3^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 *)

for(n=1, 11, for(m=1, n, print1(3^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (3^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*3)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 3*m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-3*k*x), m >= 1.
E.g.f. for m-th column: (((exp(3*x)-1)/3)^m)/m!, m >= 1.
From Peter Bala, Jan 13 2018: (Start)
Dobinski-type formulas for row polynomials R(n,x):
R(n,x) = exp(-x/3)*Sum_{i >= 0} (3*i)^n* (x/3)^i/i!;
R(n+1,x) = x*exp(-x/3)*Sum_{i >= 0} (3 + 3*i)^n* (x/3)^i/i!.
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*3^(n-k)*R(k,x).(End)

A075499 Stirling2 triangle with scaled diagonals (powers of 4).

Original entry on oeis.org

1, 4, 1, 16, 12, 1, 64, 112, 24, 1, 256, 960, 400, 40, 1, 1024, 7936, 5760, 1040, 60, 1, 4096, 64512, 77056, 22400, 2240, 84, 1, 16384, 520192, 989184, 435456, 67200, 4256, 112, 1, 65536, 4177920, 12390400, 7956480, 1779456, 169344, 7392, 144, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(4*z) - 1)*x/4) - 1
Also the inverse Bell transform of the quadruple factorial numbers 4^n*n! (A047053) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			[1]; [4,1]; [16,12,1]; ...; p(3,x) = x(16 + 12*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     4      1
*    16     12      1
*    64    112     24      1
*   256    960    400     40     1
*  1024   7936   5760   1040    60    1
*  4096  64512  77056  22400  2240   84   1
* 16384 520192 989184 435456 67200 4256 112 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000302, A016152, A019677, A075907-A075910. Row sums are A004213.

Cf. A075498, A075500.

Table[(4^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)

for(n=1, 11, for(m=1, n, print1(4^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

# uses[inverse_bell_transform from A265605]
# Adds a column 1,0,0,... at the left side of the triangle.
multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
inverse_bell_matrix(multifact_4_4, 9) # Peter Luschny, Dec 31 2015

a(n, m) = (4^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*4)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 4m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-4k*x), m >= 1.
E.g.f. for m-th column: (((exp(4x)-1)/4)^m)/m!, m >= 1.

A075500 Stirling2 triangle with scaled diagonals (powers of 5).

Original entry on oeis.org

1, 5, 1, 25, 15, 1, 125, 175, 30, 1, 625, 1875, 625, 50, 1, 3125, 19375, 11250, 1625, 75, 1, 15625, 196875, 188125, 43750, 3500, 105, 1, 78125, 1984375, 3018750, 1063125, 131250, 6650, 140, 1, 390625, 19921875

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(5*z) - 1)*x/5) - 1.

			[1]; [5,1]; [25,15,1]; ...; p(3,x) = x(25 + 15*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     5       1
*    25      15       1
*   125     175      30       1
*   625    1875     625      50      1
*  3125   19375   11250    1625     75    1
* 15625  196875  188125   43750   3500  105   1
* 78125 1984375 3018750 1063125 131250 6650 140 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000351, A016164, A075911-A075915. Row sums are A005011(n-1).

Cf. A075499, A075501.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 5^n, 9); # Peter Luschny, Jan 28 2016

Flatten[Table[5^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[5^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(5^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (5^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*5)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 5m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-5k*x), m >= 1.
E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m >= 1.

A075502 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).

Original entry on oeis.org

1, 7, 1, 49, 21, 1, 343, 343, 42, 1, 2401, 5145, 1225, 70, 1, 16807, 74431, 30870, 3185, 105, 1, 117649, 1058841, 722701, 120050, 6860, 147, 1, 823543, 14941423, 16235562, 4084101, 360150, 13034, 196, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(7*z) - 1)*x/7) - 1.

			[1]; [7,1]; [49,21,1]; ...; p(3,x) = x * (49 + 21*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*      1
*      7        1
*     49       21        1
*    343      343       42       1
*   2401     5145     1225      70      1
*  16807    74431    30870    3185    105     1
* 117649  1058841   722701  120050   6860   147   1
* 823543 14941423 16235562 4084101 360150 13034 196 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000420, A075921-A075925, A076002. Row sums are A075506.

Cf. A075501, A075503.

Flatten[Table[7^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(7^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (7^(n-m)) * stirling2(n, m).
a(n, m) = 7*m*a(n-1, m) + a(n-1, m-1), n>=m>=1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*7)^(n-m))/(m-1)! for n >= m >= 1, else 0.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-7*k*x), m >= 1.
E.g.f. for m-th column: (((exp(7*x)-1)/7)^m)/m!, m >= 1.

A075504 Stirling2 triangle with scaled diagonals (powers of 9).

Original entry on oeis.org

1, 9, 1, 81, 27, 1, 729, 567, 54, 1, 6561, 10935, 2025, 90, 1, 59049, 203391, 65610, 5265, 135, 1, 531441, 3720087, 1974861, 255150, 11340, 189, 1, 4782969, 67493007, 57041334, 11160261, 765450, 21546, 252, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(9*z) - 1)*x/9) - 1.
Row sums give A075508(n), n >= 1. The columns (without leading zeros) give A001019 (powers of 9), A076008-A076013 for m=1..7.

			[1]; [9,1]; [81,27,1]; ...; p(3,x) = x(81 + 27*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*       1
*       9        1
*      81       27        1
*     729      567       54        1
*    6561    10935     2025       90      1
*   59049   203391    65610     5265    135     1
*  531441  3720087  1974861   255150  11340   189   1
* 4782969 67493007 57041334 11160261 765450 21546 252 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A075503, A075505.

Columns 2-7 are A076008-A076013.

Flatten[Table[9^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(9^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (9^(n-m)) * stirling2(n, m).
a(n, m) = Sum_{p=0..m-1} (A075513(m, p)*((p+1)*9)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 9m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-9k*x), m >= 1.
E.g.f. for m-th column: (((exp(9x) - 1)/9)^m)/m!, m >= 1.

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