A094587 -id:A094587 - cp's OEIS frontend

A000522 Total number of ordered k-tuples (k=0..n) of distinct elements from an n-element set: a(n) = Sum_{k=0..n} n!/k!.

1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410, 9864101, 108505112, 1302061345, 16926797486, 236975164805, 3554627472076, 56874039553217, 966858672404690, 17403456103284421, 330665665962404000, 6613313319248080001, 138879579704209680022, 3055350753492612960485, 70273067330330098091156

Offset: 0

N. J. A. Sloane

Total number of permutations of all subsets of an n-set.
Also the number of one-to-one sequences that can be formed from n distinct objects.
Old name "Total number of permutations of a set with n elements", or the same with the word "arrangements", both sound too much like A000142.
Related to number of operations of addition and multiplication to evaluate a determinant of order n by cofactor expansion - see A026243.
a(n) is also the number of paths (without loops) in the complete graph on n+2 vertices starting at one vertex v1 and ending at another v2. Example: when n=2 there are 5 paths in the complete graph with 4 vertices starting at the vertex 1 and ending at the vertex 2: (12),(132),(142),(1342),(1432) so a(2) = 5. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001; comment corrected by Jonathan Coxhead, Mar 21 2003
Also row sums of Table A008279, which can be generated by taking the derivatives of x^k. For example, for y = 1*x^3, y' = 3x^2, y" = 6x, y'''= 6 so a(4) = 1 + 3 + 6 + 6 = 16. - Alford Arnold, Dec 15 1999
a(n) is the permanent of the n X n matrix with 2s on the diagonal and 1s elsewhere. - Yuval Dekel, Nov 01 2003
(A000166 + this_sequence)/2 = A009179, (A000166 - this_sequence)/2 = A009628.
Stirling transform of A006252(n-1) = [1,1,1,2,4,14,38,...] is a(n-1) = [1,2,5,16,65,...]. - Michael Somos, Mar 04 2004
Number of {12,12*,21*}- and {12,12*,2*1}-avoiding signed permutations in the hyperoctahedral group.
a(n) = b such that Integral_{x=0..1} x^n*exp(-x) dx = a-b*exp(-1). - Sébastien Dumortier, Mar 05 2005
a(n) is the number of permutations on [n+1] whose left-to-right record lows all occur at the start. Example: a(2) counts all permutations on [3] except 231 (the last entry is a record low but its predecessor is not). - David Callan, Jul 20 2005
a(n) is the number of permutations on [n+1] that avoid the (scattered) pattern 1-2-3|. The vertical bar means the "3" must occur at the end of the permutation. For example, 21354 is not counted by a(4): 234 is an offending subpermutation. - David Callan, Nov 02 2005
Number of deco polyominoes of height n+1 having no reentrant corners along the lower contour (i.e., no vertical step that is followed by a horizontal step). In other words, a(n)=A121579(n+1,0). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(1)=2 because the only deco polyominoes of height 2 are the vertical and horizontal dominoes, having no reentrant corners along their lower contours. - Emeric Deutsch, Aug 16 2006
Unreduced numerators of partial sums of the Taylor series for e. - Jonathan Sondow, Aug 18 2006
a(n) is the number of permutations on [n+1] (written in one-line notation) for which the subsequence beginning at 1 is increasing. Example: a(2)=5 counts 123, 213, 231, 312, 321. - David Callan, Oct 06 2006
a(n) is the number of permutations (written in one-line notation) on the set [n + k], k >= 1, for which the subsequence beginning at 1,2,...,k is increasing. Example: n = 2, k = 2. a(2) = 5 counts 1234, 3124, 3412, 4123, 4312. - Peter Bala, Jul 29 2014
a(n) and (1,-2,3,-4,5,-6,7,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland, Nov 01 2007
Consider the subsets of the set {1,2,3,...,n} formed by the first n integers. E.g., for n = 3 we have {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. Let the variable sbst denote a subset. For each subset sbst we determine its number of parts, that is nprts(sbst). The sum over all possible subsets is written as Sum_{sbst=subsets}. Then a(n) = Sum_{sbst=subsets} nprts(sbst)!. E.g., for n = 3 we have 1!+1!+1!+1!+2!+2!+2!+3!=16. - Thomas Wieder, Jun 17 2006
Equals row sums of triangle A158359(unsigned). - Gary W. Adamson, Mar 17 2009
Equals eigensequence of triangle A158821. - Gary W. Adamson, Mar 30 2009
For positive n, equals 1/BarnesG(n+1) times the determinant of the n X n matrix whose (i,j)-coefficient is the (i+j)th Bell number. - John M. Campbell, Oct 03 2011
a(n) is the number of n X n binary matrices with i) at most one 1 in each row and column and ii) the subset of rows that contain a 1 must also be the columns that contain a 1. Cf. A002720 where restriction ii is removed. - Geoffrey Critzer, Dec 20 2011
Number of restricted growth strings (RGS) [d(1),d(2),...,d(n)] such that d(k) <= k and d(k) <= 1 + (number of nonzero digits in prefix). The positions of nonzero digits determine the subset, and their values (decreased by 1) are the (left) inversion table (a rising factorial number) for the permutation, see example. - Joerg Arndt, Dec 09 2012
Number of a restricted growth strings (RGS) [d(0), d(1), d(2), ..., d(n)] where d(k) >= 0 and d(k) <= 1 + chg([d(0), d(1), d(2), ..., d(k-1)]) and chg(.) gives the number of changes of its argument. Replacing the function chg(.) by a function asc(.) that counts the ascents in the prefix gives A022493 (ascent sequences). - Joerg Arndt, May 10 2013
The sequence t(n) = number of i <= n such that floor(e*i!) is a square is mentioned in the abstract of Luca & Shparlinski. The values are t(n) = 0 for 0 <= n <= 2 and t(n) = 1 for at least 3 <= n <= 300. - R. J. Mathar, Jan 16 2014
a(n) = p(n,1) = q(n,1), where p and q are polynomials defined at A248664 and A248669. - Clark Kimberling, Oct 11 2014
a(n) is the number of ways at most n people can queue up at a (slow) ticket counter when one or more of the people may choose not to queue up. Note that there are C(n,k) sets of k people who quene up and k! ways to queue up. Since k can run from 0 to n, a(n) = Sum_{k=0..n} n!/(n-k)! = Sum_{k=0..n} n!/k!. For example, if n=3 and the people are A(dam), B(eth), and C(arl), a(3)=16 since there are 16 possible lineups: ABC, ACB, BAC, BCA, CAB, CBA, AB, BA, AC, CA, BC, CB, A, B, C, and empty queue. - Dennis P. Walsh, Oct 02 2015
As the row sums of A008279, Motzkin uses the abbreviated notation $n_<^\Sigma$ for a(n).
The piecewise polynomial function f defined by f(x) = a(n)*x^n/n! on each interval [ 1-1/a(n), 1-1/a(n+1) ) is continuous on [0,1) and lim_{x->1} f(x) = e. - Luc Rousseau, Oct 15 2019
a(n) is composite for 3 <= n <= 2015, but a(2016) is prime (or at least a strong pseudoprime): see Johansson link. - Robert Israel, Jul 27 2020 [a(2016) is prime, ECPP certificate generated with CM 0.4.3 and checked by factordb. - Jason H Parker, Jun 15 2025]
In general, sequences of the form a(0)=a, a(n) = n*a(n-1) + k, n>0, will have a closed form of n!*a + floor(n!*(e-1))*k. - Gary Detlefs, Oct 26 2020
From Peter Bala, Apr 03 2022: (Start)
a(2*n) is odd and a(2*n+1) is even. More generally, a(n+k) == a(n) (mod k) for all n and k. It follows that for each positive integer k, the sequence obtained by reducing a(n) modulo k is periodic, with the exact period dividing k. Various divisibility properties of the sequence follow from this; for example, a(5*n+2) == a(5*n+4) == 0 (mod 5), a(25*n+7) == a(25*n+19) == 0 (mod 25) and a(13*n+4) == a(13*n+10)== a(13*n+12) == 0 (mod 13). (End)
Number of possible ranking options on a typical ranked choice voting ballot with n candidates (allowing undervotes). - P. Christopher Staecker, May 05 2024
From Thomas Scheuerle, Dec 28 2024: (Start)
Number of decorated permutations of size n.
Number of Le-diagrams with bounding box semiperimeter n, for n > 0.
By counting over all k = 1..n and n > 0, the number of positroid cells for the totally nonnegative real Grassmannian Gr(k, n), equivalently the number of Grassmann necklaces of type (k, n). (End)

			G.f. = 1 + 2*x + 5*x^2 + 16*x^3 + 65*x^4 + 326*x^5 + 1957*x^6 + 13700*x^7 + ...
With two objects we can form 5 sequences: (), (a), (b), (a,b), (b,a), so a(2) = 5.
From _Joerg Arndt_, Dec 09 2012: (Start)
The 16 arrangements of the 3-set and their RGS (dots denote zeros) are
  [ #]       RGS        perm. of subset
  [ 1]    [ . . . ]      [  ]
  [ 2]    [ . . 1 ]      [ 3 ]
  [ 3]    [ . 1 . ]      [ 2 ]
  [ 4]    [ . 1 1 ]      [ 2 3 ]
  [ 5]    [ . 1 2 ]      [ 3 2 ]
  [ 6]    [ 1 . . ]      [ 1 ]
  [ 7]    [ 1 . 1 ]      [ 1 3 ]
  [ 8]    [ 1 . 2 ]      [ 3 1 ]
  [ 9]    [ 1 1 . ]      [ 1 2 ]
  [10]    [ 1 1 1 ]      [ 1 2 3 ]
  [11]    [ 1 1 2 ]      [ 1 3 2 ]
  [12]    [ 1 1 3 ]      [ 2 3 1 ]
  [13]    [ 1 2 . ]      [ 2 1 ]
  [14]    [ 1 2 1 ]      [ 2 1 3 ]
  [15]    [ 1 2 2 ]      [ 3 1 2 ]
  [16]    [ 1 2 3 ]      [ 3 2 1 ]
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 75, Problem 9.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 65, p. 23, Ellipses, Paris 2008.
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E11.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 16.
D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas, Freeman, San Francisco, 1974. [Annotated scans of pages 69 and 70 only]
F. Ardila, F. Rincón, and L. Williams, Positroids and non-crossing partitions, arXiv preprint arXiv:1308.2698 [math.CO], 2013.
F. Ardila, F. Rincón, and L. Williams, Positroids, non-crossing partitions, and positively oriented matroids, in FPSAC 2014, Chicago, USA; Discrete Mathematics and Theoretical Computer Science (DMTCS) Proceedings, 2014, 555-666.
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019. See Table 1, p. 6.
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7.
M. Bahrami-Taghanaki, A. R. Moghaddamfar, Nima Salehy, and Navid Salehy, Some Identities Involving Stirling Numbers Arising from Matrix Decompositions, J. Int. Seq. (2024) Vol. 24, No. 5, Art. No. 24.5.3. See p. 9.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Paul Barry, Series reversion with Jacobi and Thron continued fractions, arXiv:2107.14278 [math.NT], 2021.
Lisa Berry, Stefan Forcey, Maria Ronco, and Patrick Showers, Species substitution, graph suspension, and graded hopf algebras of painted tree polytopes, arXiv:1608.08546 [math.CO], 2016-2019.
Louis J. Billera, Sara C. Billey, and Vasu Tewari, Boolean product polynomials and Schur-positivity, arXiv:1806.02943 [math.CO], 2018.
P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials, arXiv:quant-ph/0501155, 2005.
E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35. See Table I.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Adam Brandenburger, The Strategist: Strategy from Combination, 2018.
Janusz Brzozowski and Marek Szykuła, Large Aperiodic Semigroups, arXiv preprint arXiv:1401.0157 [cs.FL], 2013.
Oscar Cabrera, Introducing loop compression for encoding de Bruijn sequences, engrXiv (2025) Art. No. 4431. See p. 16.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jean Cardinal, Arturo Merino, and Torsten Mütze, Combinatorial generation via permutation languages. IV. Elimination trees, arXiv:2106.16204 [cs.DM], 2021.
CombOS - Combinatorial Object Server, Generate partial permutations
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013.
Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson Scheme, and Generalized Derangements, International Journal of Combinatorics, Volume 2011, Article ID 539030, 29 pages.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 114.
Stefan Forcey, Aaron Lauve and Frank Sottile, Cofree compositions of coalgebras, Annals of Combinatorics 17 (1) pp. 105-130 March, 2013.
Stefan Forcey, M. Ronco, and P. Showers, Polytopes and algebras of grafted trees: Stellohedra, arXiv preprint arXiv:1608.08546 [math.CO], 2016.
Bernd Gaertner, Walter D. Jr. Morris and Leo Ruest, Unique Sink Orientations of Grids, Algorithmica, Volume 51, Number 2 / June 2008.
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885. See Ex. 2.16.
Jan Geuenich, Tilting modules for the Auslander algebra of K[x]/(xn), arXiv:1803.10707 [math.RT], 2018.
Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.
R. K. Guy, Letter to N. J. A. Sloane, 1977
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf)
Lorenz Halbeisen and Saharon Shelah, Consequences of arithmetic for set theory, The Journal of Symbolic Logic, vol. 59 (1994), pp. 30-40.
Mehdi Hassani, Counting and computing by e, arXiv:math/0606613 [math.CO], 2006.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 35
Milan Janjić, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
F. Johansson et al, MathOverflow, Is Sum_{k=1..n} (n-1)!/(k-1)! composite for n >= 4?
Michael Joswig, Max Klimm, and Sylvain Spitz, Generalized permutahedra and optimal auctions, arXiv:2108.00979 [math.MG], 2021.
Nicholas Kapoor and P. Christopher Staecker, Ahead of the Count: An Algorithm for Probabilistic Prediction of Instant Runoff (IRV) Elections, arXiv:2405.09009 [cs.CY], 2024. See p. 11.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41.
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Florian Luca and Igor E. Shparlinski, On the squarefree parts of floor(e*n!), Glasgow Math. J., 49 (2007), 391-403.
Thibault Manneville and Vincent Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
Emanuele Munarini, Combinatorial Identities for the Tricomi Polynomials, J. Int. Seq., Vol. 23 (2020), Article 20.9.4.
Emanuele Munarini, Two-Parameter Identities for q-Appell Polynomials, Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.1.
R. Ondrejka, Letter to N. J. A. Sloane, May 15 1976
Jason H Parker, Certificate for a(2016), generated with CM 0.4.4dev
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70. [Annotated scanned copy]
Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641. arXiv:0704.1282 [math.HO], 2007-2010.
Jonathan Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection
Jonathan Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, arXiv:0709.0671 [math.NT], 2006-2009; Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Eric Weisstein's World of Mathematics, Binomial Sums
Index entries for sequences related to logarithmic numbers
Index entries for related partition-counting sequences
Index entries for sequences related to factorial numbers

Cf. A000166, A002627, A006231, A064383, A064384, A008290, A010844, A010845, A014508, A038159, A054091, A058006, A072453, A072456, A073591, A082030, A095000, A095177, A108625, A121579, A124779, A142992, A143007, A158359, A158821, A195254, A222637-A222639, A038155, A000217.
Average of n-th row of triangle in A068424 [Corrected by N. J. A. Sloane, Feb 29 2024].
Row sums of A008279 and A094816.
First differences give A001339. Second differences give A001340.
Partial sums are in A001338, A002104.
A row of the array in A144502.
See also A370973, Nearest integer to e*n!.

import Data.List (subsequences, permutations)
a000522 = length . choices . enumFromTo 1 where
choices = concat . map permutations . subsequences
-- Reinhard Zumkeller, Feb 21 2012, Oct 25 2010

[1] cat [n eq 1 select (n+1) else n*Self(n-1)+1: n in [1..25]]; // Vincenzo Librandi, Feb 15 2015

a(n):= exp(1)*int(x^n*exp(-x)*Heaviside(x-1), x=0..infinity); # Karol A. Penson, Oct 01 2001
A000522 := n->add(n!/k!,k=0..n);
G(x):=exp(x)/(1-x): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20);
# Zerinvary Lajos, Apr 03 2009
G:=exp(z)/(1-z): Gser:=series(G,z=0,21):
for n from 0 to 20 do a(n):=n!*coeff(Gser,z,n): end do
# Paul Weisenhorn, May 30 2010
k := 1; series(hypergeom([1,k],[],x/(1-x))/(1-x), x=0, 20); # Mark van Hoeij, Nov 07 2011
# one more Maple program:
a:= proc(n) option remember;
      `if`(n<0, 0, 1+n*a(n-1))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 13 2019
seq(simplify(KummerU(-n, -n, 1)), n = 0..23); # Peter Luschny, May 10 2022

Table[FunctionExpand[Gamma[n + 1, 1]*E], {n, 0, 24}]
nn = 20; Accumulate[Table[1/k!, {k, 0, nn}]] Range[0, nn]! (* Jan Mangaldan, Apr 21 2013 *)
FoldList[#1*#2 + #2 &, 0, Range@ 23] + 1 (* or *)
f[n_] := Floor[E*n!]; f[0] = 1; Array[f, 20, 0] (* Robert G. Wilson v, Feb 13 2015 *)
RecurrenceTable[{a[n + 1] == (n + 1) a[n] + 1, a[0] == 1}, a, {n, 0, 12}] (* Emanuele Munarini, Apr 27 2017 *)
nxt[{n_,a_}]:={n+1,a(n+1)+1}; NestList[nxt,{0,1},30][[All,2]] (* Harvey P. Dale, Jan 29 2023 *)

a(n) := if n=0 then 1 else n*a(n-1)+1; makelist(a(n),n,0,12); /* Emanuele Munarini, Apr 27 2017 */

{a(n) = local(A); if( n<0, 0, A = vector(n+1); A[1]=1; for(k=1, n, A[k+1] = k*A[k] + 1); A[n+1])}; /* Michael Somos, Jul 01 2004 */

{a(n) = if( n<0, 0, n! * polcoeff( exp( x +x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Mar 06 2004 */

a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x)^2+x^2*deriv(A)/(1-x)); polcoeff(A, n) \\ Paul D. Hanna, Sep 03 2008

{a(n)=local(X=x+x*O(x^n));polcoeff(sum(m=0,n,(m+2)^m*x^m/(1+(m+1)*X)^(m+1)),n)} /* Paul D. Hanna */

a(n)=sum(k=0,n,binomial(n,k)*k!); \\ Joerg Arndt, Dec 14 2014

# program adapted from Alois P. Heinz's Maple code in A022493
@CachedFunction
def b(n, i, t):
    if n <= 1:
        return 1
    return sum(b(n - 1, j, t + (j == i)) for j in range(t + 2))
def a(n):
    return b(n, 0, 0)
v000522 = [a(n) for n in range(33)]
print(v000522)
# Joerg Arndt, May 11 2013

a(n) = n*a(n-1) + 1, a(0) = 1.
a(n) = A007526(n-1) + 1.
a(n) = A061354(n)*A093101(n).
a(n) = n! * Sum_{k=0..n} 1/k! = n! * (e - Sum_{k>=n+1} 1/k!). - Michael Somos, Mar 26 1999
a(0) = 1; for n > 0, a(n) = floor(e*n!).
E.g.f.: exp(x)/(1-x).
a(n) = 1 + Sum_{n>=k>=j>=0} (k-j+1)*k!/j! = a(n-1) + A001339(n-1) = A007526(n) + 1. Binomial transformation of n!, i.e., A000142. - Henry Bottomley, Jun 04 2001
a(n) = floor(2/(n+1))*A009578(n+1)-1. - Emeric Deutsch, Oct 24 2001
Integral representation as n-th moment of a nonnegative function on a positive half-axis: a(n) = e*Integral_{x>=0} x^n*e^(-x)*Heaviside(x-1) dx. - Karol A. Penson, Oct 01 2001
Formula, in Mathematica notation: Special values of Laguerre polynomials, a(n)=(-1)^n*n!*LaguerreL[n, -1-n, 1], n=1, 2, ... . This relation cannot be checked by Maple, as it appears that Maple does not incorporate Laguerre polynomials with second index equal to negative integers. It does check with Mathematica. - Karol A. Penson and Pawel Blasiak ( blasiak(AT)lptl.jussieu.fr), Feb 13 2004
G.f.: Sum_{k>=0} k!*x^k/(1-x)^(k+1). a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*k^(n-k)*(k+1)^k. - Vladeta Jovovic, Aug 18 2002
a(n) = Sum_{k=0..n} A008290(n, k)*2^k. - Philippe Deléham, Dec 12 2003
a(n) = Sum_{k=0..n} A046716(n, k). - Philippe Deléham, Jun 12 2004
a(n) = e*Gamma(n+1,1) where Gamma(z,t) = Integral_{x>=t} e^(-x)*x^(z-1) dx is incomplete gamma function. - Michael Somos, Jul 01 2004
a(n) = Sum_{k=0..n} P(n, k). - Ross La Haye, Aug 28 2005
Given g.f. A(x), then g.f. A059115 = A(x/(1-x)). - Michael Somos, Aug 03 2006
a(n) = 1 + n + n*(n-1) + n*(n-1)*(n-2) + ... + n!. - Jonathan Sondow, Aug 18 2006
a(n) = Sum_{k=0..n} binomial(n,k) * k!; interpretation: for all k-subsets (sum), choose a subset (binomial(n,k)), and permutation of subset (k!). - Joerg Arndt, Dec 09 2012
a(n) = Integral_{x>=0} (x+1)^n*e^(-x) dx. - Gerald McGarvey, Oct 19 2006
a(n) = Sum_{k=0..n} A094816(n, k), n>=0 (row sums of Poisson-Charlier coefficient matrix). - N. J. A. Sloane, Nov 10 2007
From Tom Copeland, Nov 01 2007, Dec 10 2007: (Start)
Act on 1/(1-x) with 1/(1-xDx) = Sum_{j>=0} (xDx)^j = Sum_{j>=0} x^j*D^j*x^j = Sum_{j>=0} j!*x^j*L(j,-:xD:,0) where Lag(n,x,0) are the Laguerre polynomials of order 0, D the derivative w.r.t. x and (:xD:)^j = x^j*D^j. Truncating the operator series at the j = n term gives an o.g.f. for a(0) through a(n) consistent with Jovovic's.
These results and those of Penson and Blasiak, Arnold, Bottomley and Deleham are related by the operator A094587 (the reverse of A008279), which is the umbral equivalent of n!*Lag[n,(.)!*Lag[.,x,-1],0] = (1-D)^(-1) x^n = (-1)^n * n! Lag(n,x,-1-n) = Sum_{j=0..n} binomial(n,j)*j!*x^(n-j) = Sum_{j=0..n} (n!/j!)*x^j. Umbral substitution of b(.) for x and then letting b(n)=1 for all n connects the results. See A132013 (the inverse of A094587) for a connection between these operations and 1/(1-xDx).
(End)
From Peter Bala, Jul 15 2008: (Start)
a(n) = n!*e - 1/(n + 1/(n+1 + 2/(n+2 + 3/(n+3 + ...)))).
Asymptotic result (Ramanujan): n!*e - a(n) ~ 1/n - 1/n^3 + 1/n^4 + 2/n^5 - 9/n^6 + ..., where the sequence [1,0,-1,1,2,-9,...] = [(-1)^k*A000587(k)], for k>=1.
a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). For fixed k, define the derived sequence a_k(n) = (a(n+k)-a(k))/n, n = 1,2,3,... . Then a_k(n) is also a difference divisibility sequence.
For example, the derived sequence a_0(n) is just a(n-1). The set of integer sequences satisfying the difference divisibility property forms a ring with term-wise operations of addition and multiplication.
Recurrence relations: a(0) = 1, a(n) = (n-1)*(a(n-1) + a(n-2)) + 2, for n >= 1. a(0) = 1, a(1) = 2, D-finite with recurrence: a(n) = (n+1)*a(n-1) - (n-1)*a(n-2) for n >= 2. The sequence b(n) := n! satisfies the latter recurrence with the initial conditions b(0) = 1, b(1) = 1. This leads to the finite continued fraction expansion a(n)/n! = 1/(1-1/(2-1/(3-2/(4-...-(n-1)/(n+1))))), n >= 2.
Limit_{n->oo} a(n)/n! = e = 1/(1-1/(2-1/(3-2/(4-...-n/((n+2)-...))))). This is the particular case m = 0 of the general result m!/e - d_m = (-1)^(m+1) *(1/(m+2 -1/(m+3 -2/(m+4 -3/(m+5 -...))))), where d_m denotes the m-th derangement number A000166(m).
For sequences satisfying the more general recurrence a(n) = (n+1+r)*a(n-1) - (n-1)*a(n-2), which yield series acceleration formulas for e/r! that involve the Poisson-Charlier polynomials c_r(-n;-1), refer to A001339 (r=1), A082030 (r=2), A095000 (r=3) and A095177 (r=4).
For the corresponding results for the constants log(2), zeta(2) and zeta(3) refer to A142992, A108625 and A143007 respectively.
(End)
G.f. satisfies: A(x) = 1/(1-x)^2 + x^2*A'(x)/(1-x). - Paul D. Hanna, Sep 03 2008
From Paul Barry, Nov 27 2009: (Start)
G.f.: 1/(1-2*x-x^2/(1-4*x-4*x^2/(1-6*x-9*x^2/(1-8*x-16*x^2/(1-10*x-25*x^2/(1-... (continued fraction);
G.f.: 1/(1-x-x/(1-x/(1-x-2*x/(1-2*x/(1-x-3*x/(1-3*x/(1-x-4*x/(1-4*x/(1-x-5*x/(1-5*x/(1-... (continued fraction).
(End)
O.g.f.: Sum_{n>=0} (n+2)^n*x^n/(1 + (n+1)*x)^(n+1). - Paul D. Hanna, Sep 19 2011
G.f. hypergeom([1,k],[],x/(1-x))/(1-x), for k=1,2,...,9 is the generating function for A000522, A001339, A082030, A095000, A095177, A096307, A096341, A095722, and A095740. - Mark van Hoeij, Nov 07 2011
G.f.: 1/U(0) where U(k) = 1 - x - x*(k+1)/(1 - x*(k+1)/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2012
E.g.f.: 1/U(0) where U(k) = 1 - x/(1 - 1/(1 + (k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 16 2012
G.f.: 1/(1-x)/Q(0), where Q(k) = 1 - x/(1-x)*(k+1)/(1 - x/(1-x)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
G.f.: 2/(1-x)/G(0), where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+3) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: (B(x)+ 1)/(2-2*x) = Q(0)/(2-2*x), where B(x) be g.f. A006183, Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 08 2013
G.f.: 1/Q(0), where Q(k) = 1 - 2*x*(k+1) - x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
E.g.f.: e^x/(1-x) = (1 - 12*x/(Q(0) + 6*x - 3*x^2))/(1-x), where Q(k) = 2*(4*k+1)*(32*k^2 + 16*k + x^2 - 6) - x^4*(4*k-1)*(4*k+7)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
G.f.: conjecture: T(0)/(1-2*x), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
0 = a(n)*(+a(n+1) - 3*a(n+2) + a(n+3)) + a(n+1)*(+a(n+1) - a(n+3)) + a(n+2)*(+a(n+2)) for all n>=0. - Michael Somos, Jul 04 2014
From Peter Bala, Jul 29 2014: (Start)
a(n) = F(n), where the function F(x) := Integral_{0..infinity} e^(-u)*(1 + u)^x du smoothly interpolates this sequence to all real values of x. Note that F(-1) = G and for n = 2,3,... we have F(-n) = (-1)^n/(n-1)! *( A058006(n-2) - G ), where G = 0.5963473623... denotes Gompertz's constant - see A073003.
a(n) = n!*e - e*( Sum_{k >= 0} (-1)^k/((n + k + 1)*k!) ).
(End)
a(n) = hypergeometric_U(1, n+2, 1). - Peter Luschny, Nov 26 2014
a(n) ~ exp(1-n)*n^(n-1/2)*sqrt(2*Pi). - Vladimir Reshetnikov, Oct 27 2015
a(n) = A038155(n+2)/A000217(n+1). - Anton Zakharov, Sep 08 2016
a(n) = round(exp(1)*n!), n > 1 - Simon Plouffe, Jul 28 2020
a(n) = KummerU(-n, -n, 1). - Peter Luschny, May 10 2022
a(n) = (e/(2*Pi))*Integral_{x=-oo..oo} (n+1+i*x)!/(1+i*x) dx. - David Ulgenes, Apr 18 2023
Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * a(i) = n!. - Werner Schulte, Apr 03 2024

Additional comments from Michael Somos

A001710 Order of alternating group A_n, or number of even permutations of n letters.

Original entry on oeis.org

1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000

Offset: 0

N. J. A. Sloane

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001
a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad Brewbaker, Jan 31 2003
a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad Brewbaker, Jan 31 2003
Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch, Aug 28 2004
Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini, Oct 15 2004
The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - Jon Perry, Sep 20 2008
Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - Gary W. Adamson, Dec 25 2008
First column of A092582. - Mats Granvik, Feb 08 2009
The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009
For n>1: a(n) = A173333(n,2). - Reinhard Zumkeller, Feb 19 2010
Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - Gary W. Adamson, Aug 02 2010
For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - Geoffrey Critzer, Feb 16 2011.
The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011
Also [1, 1] together with the row sums of triangle A162608. - Omar E. Pol, Mar 09 2012
a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11,  n=3: 112, n=4: 1123, 1132, 1213,  n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012
For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - Stanislav Sykora, May 10 2014
In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - Antti Karttunen, Dec 19 2015
Also (by definition) the independence number of the n-transposition graph. - Eric W. Weisstein, May 21 2017
Number of permutations of n letters containing an even number of even cycles. - Michael Somos, Jul 11 2018
Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - Gus Wiseman, Oct 20 2018
For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - Noah A Rosenberg, Feb 11 2019
Also the clique covering number of the n-Bruhat graph. - Eric W. Weisstein, Apr 19 2019
a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020
For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - Ivan N. Ianakiev, Mar 11 2020
For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - Dennis P. Walsh, May 28 2020

			G.f. = 1 + x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 360*x^6 + 2520*x^7 + ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 87-8, 20. (a), c_n^e(t=1).
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), Article 00.1.5.
Camille Combe and Samuele Giraudo, Cliff operads: a hierarchy of operads on words, arXiv:2106.14552 [math.CO], 2021.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 7.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 262.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Shirali Kadyrov and Farukh Mashurov, Generalized continued fraction expansions for Pi and E, arXiv:1912.03214 [math.NT], 2019.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Xah Lee, Combinatorics: Loop in n points.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
S-Z Song, S-G Hwang, S-H Rim, and G-S Cheon, Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Alternating Group.
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Circular Permutation.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Even Permutation.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Odd Permutation.
Eric Weisstein's World of Mathematics, Transposition Graph.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.
Index entries for sequences related to factorial base representation.
Index entries for sequences related to factorial numbers.
Index entries for sequences related to groups.
Index to divisibility sequences.

Cf. A000142, A000153, A000255, A001147, A001286, A001720, A002135, A002260, A007623, A007717, A049444, A049459, A093468, A094587, A094638, A138533, A153880, A173333, A213936, A215771, A319225, A319226, A320655.
a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A161739 (q(n) sequence).
Bisections are A002674 and A085990 (essentially).
Row 3 of A265609 (essentially).
Row sums of A307429.

[1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // Arkadiusz Wesolowski, May 17 2014

seq(mul(k, k=3..n), n=0..20); # Zerinvary Lajos, Sep 14 2007

a[n_]:= If[n > 2, n!/2, 1]; Array[a, 21, 0]
a[n_]:= If[n<3, 1, n*a[n-1]]; Array[a, 21, 0]; (* Robert G. Wilson v, Apr 16 2011 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[(2-x^2)/(2-2x), {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Sinh[-Log[1-x]], {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
Numerator[Range[0, 20]!/2] (* Eric W. Weisstein, May 21 2017 *)
Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* Eric W. Weisstein, May 21 2017 *)

PARI
```
{a(n) = if( n<2, n>=0, n!/2)};
```

a(n)=polcoeff(1+x*sum(m=0,n,m^m*x^m/(1+m*x+x*O(x^n))^m),n) \\ Paul D. Hanna

A001710=n->n!\2+(n<2) \\ M. F. Hasler, Dec 01 2013

from math import factorial
def A001710(n): return factorial(n)>>1 if n > 1 else 1 # Chai Wah Wu, Feb 14 2023

def A001710(n): return (factorial(n) +int(n<2))//2
[A001710(n) for n in range(31)] # G. C. Greubel, Sep 28 2024

;; Using memoization-macro definec for which an implementation can be found in http://oeis.org/wiki/Memoization
(definec (A001710 n) (cond ((<= n 2) 1) (else (* n (A001710 (- n 1))))))
;; Antti Karttunen, Dec 19 2015

a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
D-finite with recurrence: a(0) = a(1) = a(2) = 1; a(n) = n*a(n-1) for n>2. - Chad Brewbaker, Jan 31 2003 [Corrected by N. J. A. Sloane, Jul 25 2008]
a(0) = 0, a(1) = 1; a(n) = Sum_{k=1..n-1} k*a(k). - Amarnath Murthy, Oct 29 2002
Stirling transform of a(n+1) = [1, 3, 12, 160, ...] is A083410(n) = [1, 4, 22, 154, ...]. - Michael Somos, Mar 04 2004
First Eulerian transform of A000027. See A000142 for definition of FET. - Ross La Haye, Feb 14 2005
From Paul Barry, Apr 18 2005: (Start)
a(n) = 0^n + Sum_{k=0..n} (-1)^(n-k-1)*T(n-1, k)*cos(Pi*(n-k-1)/2)^2.
T(n,k) = abs(A008276(n, k)). (End)
E.g.f.: (2 - x^2)/(2 - 2*x).
E.g.f. of a(n+2), n>=0, is 1/(1-x)^3.
E.g.f.: 1 + sinh(log(1/(1-x))). - Geoffrey Critzer, Dec 12 2010
a(n+1) = (-1)^n * A136656(n,1), n>=1.
a(n) = n!/2 for n>=2 (proof from the e.g.f). - Wolfdieter Lang, Apr 30 2010
a(n) = (n-2)! * t(n-1), n>1, where t(n) is the n-th triangular number (A000217). - Gary Detlefs, May 21 2010
a(n) = ( A000254(n) - 2* A001711(n-3) )/3, n>2. - Gary Detlefs, May 24 2010
O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n. - Paul D. Hanna, Sep 13 2011
a(n) = if n < 2 then 1, otherwise Pochhammer(n,n)/binomial(2*n,n). - Peter Luschny, Nov 07 2011
a(n) = Sum_{k=0..floor(n/2)} s(n,n-2*k) where s(n,k) are Stirling number of the first kind, A048994. - Mircea Merca, Apr 07 2012
a(n-1), n>=3, is M_1([2,1^(n-2)])/n = (n-1)!/2, with the M_1 multinomial numbers for the given n-1 part partition of n. See the second to last entry in line n>=3 of A036038, and the above necklace comment by W. Lang. - Wolfdieter Lang, Jun 26 2012
G.f.: A(x) = 1 + x + x^2/(G(0)-2*x) where G(k) = 1 - (k+1)*x/(1 - x*(k+3)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012.
G.f.: 1 + x + (Q(0)-1)*x^2/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+2)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + (x*Q(x)-x^2)/(2*(sqrt(x)+x)), where Q(x) = Sum_{n>=0} (n+1)!*x^n*sqrt(x)*(sqrt(x) + x*(n+2)). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x/2 + (Q(0)-1)*x/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+1)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: 1+x + x^2*W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
From Antti Karttunen, Dec 19 2015: (Start)
a(0)=a(1)=1; after which, for even n: a(n) = (n/2) * (n-1)!, and for odd n: a(n) = (n-1)/2 * ((n-1)! + (n-2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proved by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.]
For n >= 1, a(2*n+1) = a(2*n) + A153880(a(2*n)). [Follows from above.] (End)
Inverse Stirling transform of a(n) is (-1)^(n-1)*A009566(n). - Anton Zakharov, Aug 07 2016
a(n) ~ sqrt(Pi/2)*n^(n+1/2)/exp(n). - Ilya Gutkovskiy, Aug 07 2016
a(n) = A006595(n-1)*n/A000124(n) for n>=2. - Anton Zakharov, Aug 23 2016
a(n) = A001563(n-1) - A001286(n-1) for n>=2. - Anton Zakharov, Sep 23 2016
From Peter Bala, May 24 2017: (Start)
The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (x - 1)*A(x) + 1 - x^2 = 0.
G.f.: A(x) = 1 + x + x^2/(1 - 3*x/(1 - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1  - ... - (n + 2)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1 + x + x^2/(1 - 2*x - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-2)) / 2 = x - 3*x^2/2! + 12*x^3/3! - ..., an e.g.f. for the signed sequence here (n!/2!), ignoring the first two terms, is the compositional inverse of G(x) = (1 - 2*x)^(-1/2) - 1 = x + 3*x^2/2! + 15*x^3/3! + ..., an e.g.f. for A001147. Cf. A094638. H(x) is the e.g.f. for the sequence (-1)^m * m!/2 for m = 2,3,4,... . Cf. A001715 for n!/3! and A001720 for n!/4!. Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 2*(e-1).
Sum_{n>=0} (-1)^n/a(n) = 2/e. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001

Further terms from Simone Severini, Oct 15 2004

A000254 Unsigned Stirling numbers of first kind, s(n+1,2): a(n+1) = (n+1)*a(n) + n!.

Original entry on oeis.org

0, 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576, 10628640, 120543840, 1486442880, 19802759040, 283465647360, 4339163001600, 70734282393600, 1223405590579200, 22376988058521600, 431565146817638400, 8752948036761600000, 186244810780170240000

Offset: 0

N. J. A. Sloane

Number of permutations of n+1 elements with exactly two cycles.
Number of cycles in all permutations of [n]. Example: a(3) = 11 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) have 11 cycles altogether. - Emeric Deutsch, Aug 12 2004
Row sums of A094310: In the symmetric group S_n, each permutation factors into k independent cycles; a(n) = sum k over S_n. - Harley Flanders (harley(AT)umich.edu), Jun 28 2004
The sum of the top levels of the last column over all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=3 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, the levels of their last columns being 2 and 1, respectively. - Emeric Deutsch, Aug 12 2006
a(n) is divisible by n for all composite n >= 6. a(2*n) is divisible by 2*n + 1. - Leroy Quet, May 20 2007
For n >= 2 the determinant of the n-1 X n-1 matrix M(i,j) = i + 2 for i = j and 1 otherwise (i,j = 1..n-1). E.g., for n = 3 the determinant of [(3, 1), (1, 4)]. See 53rd Putnam Examination, 1992, Problem B5. - Franz Vrabec, Jan 13 2008, Mar 26 2008
The numerator of the fraction when we sum (without simplification) the terms in the harmonic sequence. (1 + 1/2 = 2/2 + 1/2 = 3/2; 3/2 + 1/3 = 9/6 + 2/6 = 11/6; 11/6 + 1/4 = 44/24 + 6/24 = 50/24;...). The denominator of this fraction is n!*A000142. - Eric Desbiaux, Jan 07 2009
The asymptotic expansion of the higher order exponential integral E(x,m=2,n=1) ~ exp(-x)/x^2*(1 - 3/x + 11/x^2 - 50/x^3 + 274/x^4 - 1764/x^5 + 13068/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009
a(n) is the number of permutations of [n+1] containing exactly 2 cycles. Example: a(2) = 3 because the permutations (1)(23), (12)(3), (13)(2) are the only permutations of [3] with exactly 2 cycles. - Tom Woodward (twoodward(AT)macalester.edu), Nov 12 2009
It appears that, with the exception of n= 4, a(n) mod n = 0 if n is composite and = n-1 if n is prime. - Gary Detlefs, Sep 11 2010
a(n) is a multiple of A025527(n). - Charles R Greathouse IV, Oct 16 2012
Numerator of harmonic number H(n) = Sum_{i=1..n} 1/i when not reduced. See A001008 (Wolstenholme numbers) for the reduced numerators. - Rahul Jha, Feb 18 2015
The Stirling transform of this sequence is A222058(n) (Harmonic-geometric numbers). - Anton Zakharov, Aug 07 2016
a(n) is the (n-1)-st elementary symmetric function of the first n numbers. - Anton Zakharov, Nov 02 2016
The n-th iterated integral of log(x) is x^n * (n! * log(x) - a(n))/(n!)^2 + a polynomial of degree n-1 with arbitrary coefficients. This can be proven using the recurrence relation a(n) = (n-1)! + n*a(n-1). - Mohsen Maesumi, Oct 31 2018
Primes p such that p^3 | a(p-1) are the Wolstenholme primes A088164. - Amiram Eldar and Thomas Ordowski, Aug 08 2019
Total number of left-to-right maxima (or minima) in all permutations of [n]. a(3) = 11 = 3+2+2+2+1+1: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21. - Alois P. Heinz, Aug 01 2020

			(1-x)^-1 * (-log(1-x)) = x + 3/2*x^2 + 11/6*x^3 + 25/12*x^4 + ...
G.f. = x + x^2 + 5*x^3 + 14*x^4 + 94*x^5 + 444*x^6 + 3828*x^7 + 25584*x^8 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identities 186-190.
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, 1986, see page 2. MR0863284 (89d:41049)
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
Shanzhen Gao, Permutations with Restricted Structure (in preparation).
K. Javorszky, Natural Orders: De Ordinibus Naturalibus, 2016, ISBN 978-3-99057-139-2.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..449 (terms 0..100 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159 (1996), 29-42.
J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, preprint, 2016.
FindStat - Combinatorial Statistic Finder, The number of cycles in the cycle decomposition of a permutation.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 31.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv:1201.1323 [math.CO], 2012.
Sergey Kitaev and Jeffrey Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012), #12.4.7.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.
J. Riordan, Letter of 04/11/74.
John A. Rochowicz Jr., Harmonic Numbers: Insights, Approximations and Applications, Spreadsheets in Education (eJSiE), 8(2) (2015), Article 4.
N. A. Rosenberg, Informativeness of genetic markers for inference of ancestry, American Journal of Human Genetics 73 (2003), 1402-1422.
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), #10.6.7, Section 4.3.2.
J. Scholes, 53rd Putnam 1992, Problem B5.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles. (Annotated scans of some selected pages)
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 5.

Cf. A000399, A000454, A000482, A001233, A001234, A243569, A243570.
Cf. A000774, A004041, A024167, A046674, A049034, A008275.
Cf. A021009, A081358, A092691, A094587, A151881, A121633.
With signs: A081048.
Column 1 in triangle A008969.
Row sums of A136662.

a:=[]; for n in [1..22] do a:=a cat [Abs(StirlingFirst(n,2))]; end for; a; // Marius A. Burtea, Jan 01 2020

A000254 := proc(n) option remember; if n<=1 then n else n*A000254(n-1)+(n-1)!; fi; end: seq(A000254(n),n=0..21);
a := n -> add(n!/k, k=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jan 22 2008

Table[ (PolyGamma[ m ]+EulerGamma) (m-1)!, {m, 1, 24} ] (* Wouter Meeussen *)
Table[ n!*HarmonicNumber[n], {n, 0, 19}] (* Robert G. Wilson v, May 21 2005 *)
Table[Sum[1/i,{i,1,n}]/Product[1/i,{i,1,n}],{n,1,30}] (* Alexander Adamchuk, Jul 11 2006 *)
Abs[StirlingS1[Range[20],2]] (* Harvey P. Dale, Aug 16 2011 *)
Table[Gamma'[n + 1] /. EulerGamma -> 0, {n, 0, 30}] (* Li Han, Feb 14 2024*)

a(n):=(-1)^(n+1)/2*(n+1)*sum(k*bern(k-1)*stirling1(n,k),k,1,n); /* Vladimir Kruchinin, Nov 20 2016 */

A000254 := proc(n) begin n*A000254(n-1)+fact(n-1) end_proc: A000254(1) := 1:

{a(n) = if( n<0, 0, (n+1)! / 2 * sum( k=1, n, 1 / k / (n+1-k)))} /* Michael Somos, Feb 05 2004 */

[stirling_number1(i, 2) for i in range(1, 22)]  # Zerinvary Lajos, Jun 27 2008

Let P(n,X) = (X+1)*(X+2)*(X+3)*...*(X+n); then a(n) is the coefficient of X; or a(n) = P'(n,0). - Benoit Cloitre, May 09 2002
Sum_{k > 0} a(k) * x^k/ k!^2 = exp(x) *(Sum_{k>0} (-1)^(k+1) * x^k / (k * k!)). - Michael Somos, Mar 24 2004; corrected by Warren D. Smith, Feb 12 2006
a(n) is the coefficient of x^(n+2) in (-log(1-x))^2, multiplied by (n+2)!/2.
a(n) = n! * Sum_{i=1..n} 1/i = n! * H(n), where H(n) = A001008(n)/A002805(n) is the n-th harmonic number.
a(n) ~ 2^(1/2)*Pi^(1/2)*log(n)*n^(1/2)*e^-n*n^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: log(1 - x) / (x-1). (= (log(1 - x))^2 / 2 if offset 1). - Michael Somos, Feb 05 2004
D-finite with recurrence: a(n) = a(n-1) * (2*n - 1) - a(n-2) * (n - 1)^2, if n > 1. - Michael Somos, Mar 24 2004
a(n) = A081358(n)+A092691(n). - Emeric Deutsch, Aug 12 2004
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k. - Vladeta Jovovic, Jan 29 2005
p^2 divides a(p-1) for prime p > 3. a(n) = (Sum_{i=1..n} 1/i) / Product_{i=1..n} 1/i. - Alexander Adamchuk, Jul 11 2006
a(n) = 3* A001710(n) + 2* A001711(n-3) for n > 2; e.g., 11 = 3*3 + 2*1, 50 = 3*12 + 2*7, 274 = 3*60 + 2*47, ... - Gary Detlefs, May 24 2010
a(n) = A138772(n+1) - A159324(n). - Gary Detlefs, Jul 05 2010
a(n) = A121633(n) + A002672(n). - Gary Detlefs, Jul 18 2010
a(n+1) = Sum_{i=1..floor((n-1)/2)} n!/((n-i)*i) + Sum_{i=ceiling(n/2)..floor(n/2)} n!/(2*(n-i)*i). - Shanzhen Gao, Sep 14 2010
From Gary Detlefs, Sep 11 2010: (Start)
a(n) = (a(n-1)*(n^2 - 2*n + 1) + (n + 1)!)/(n - 1) for n > 2.
It appears that, with the exception of n = 2, (a(n+1)^2 - a(n)^2) mod n^2 = 0 if n is composite and 4*n if n is prime.
It appears that, with the exception of n = 2, (a(n+1)^3 - a(n)^2) mod n = 0 if n is composite and n - 2 if n is prime.
It appears that, with the exception of n = 2, (a(n)^2 + a(n+1)^2) mod n = 0 if n is composite and = 2 if n is prime. (End)
a(n) = Integral_{x=0..oo} (x^n - n!)*log(x)*exp(-x) dx. - Groux Roland, Mar 28 2011
a(n) = 3*n!/2 + 2*(n-2)!*Sum_{k=0..n-3} binomial(k+2,2)/(n-2-k) for n >= 2. - Gary Detlefs, Sep 02 2011
a(n)/(n-1)! = ml(n) = n*ml(n-1)/(n-1) + 1 for n > 1, where ml(n) is the average number of random draws from an n-set with replacement until the total set has been observed. G.f. of ml: x*(1 - log(1 - x))/(1 - x)^2. - Paul Weisenhorn, Nov 18 2011
a(n) = det(|S(i+2, j+1)|, 1 <= i,j <= n-2), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
E.g.f.: x/(1 - x)*E(0)/2, where E(k) = 2 + E(k+1)*x*(k + 1)/(k + 2). - Sergei N. Gladkovskii, Jun 01 2013 [Edited by Michael Somos, Nov 28 2013]
0 = a(n) * (a(n+4) - 6*a(n+3) + 7*a(n+2) - a(n+1)) - a(n+1) * (4*a(n+3) - 6*a(n+2) + a(n+1)) + 3*a(n+2)^2 unless n=0. - Michael Somos, Nov 28 2013
For a simple way to calculate the sequence, multiply n! by the integral from 0 to 1 of (1 - x^n)/(1 - x) dx. - Rahul Jha, Feb 18 2015
From Ilya Gutkovskiy, Aug 07 2016: (Start)
Inverse binomial transform of A073596.
a(n) ~ sqrt(2*Pi*n) * n^n * (log(n) + gamma)/exp(n), where gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = ((-1)^(n+1)/2*(n+1))*Sum_{k=1..n} k*Bernoulli(k-1)*Stirling1(n,k). - Vladimir Kruchinin, Nov 20 2016
a(n) = (n)! * (digamma(n+1) + gamma), where gamma is the Euler-Mascheroni constant A001620. - Pedro Caceres, Mar 10 2018
From Andy Nicol, Oct 21 2021: (Start)
Gamma'(x) = a(x-1) - (x-1)!*gamma, where Gamma'(x) is the derivative of the gamma function at positive integers and gamma is the Euler-Mascheroni constant. E.g.:
Gamma'(1) = -gamma, Gamma'(2) = 1-gamma, Gamma'(3) = 3-2*gamma,
Gamma'(22) = 186244810780170240000 - 51090942171709440000*gamma. (End)
From Peter Bala, Feb 03 2022: (Start)
The following are all conjectural:
E.g.f.: for nonzero m, (1/m)*Sum_{n >= 1} (-1)^(n+1)*(1/n)*binomial(m*n,n)* x^n/(1 - x)^(m*n+1) = x + 3*x^2/2! + 11*x^3/3! + 50*x^4/4! + ....
For nonzero m, a(n) = (1/m)*n!*Sum_{k = 1..n} (-1)^(k+1)*(1/k)*binomial(m*k,k)* binomial(n+(m-1)*k,n-k).
a(n)^2 = (1/2)*n!^2*Sum_{k = 1..n} (-1)^(k+1)*(1/k^2)*binomial(n,k)* binomial(n+k,k). (End)
From Mélika Tebni, Jun 20 2022: (Start)
a(n) = -Sum_{k=0..n} k!*A021009(n, k+1).
a(n) = Sum_{k=0..n} k!*A094587(n, k+1). (End)
a(n) = n! * 1/(1 - 1^2/(3 - 2^2/(5 - 3^2/(7 - ... - (n - 1)^2/((2*n - 1)))))). - Peter Bala, Mar 16 2024

A008279 Triangle T(n,k) = n!/(n-k)! (0 <= k <= n) read by rows, giving number of permutations of n things k at a time.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 12, 24, 24, 1, 5, 20, 60, 120, 120, 1, 6, 30, 120, 360, 720, 720, 1, 7, 42, 210, 840, 2520, 5040, 5040, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 1, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 362880

Offset: 0

N. J. A. Sloane

Also called permutation coefficients.
Also falling factorials triangle A068424 with column a(n,0)=1 and row a(0,1)=1 otherwise a(0,k)=0, added. - Wolfdieter Lang, Nov 07 2003
The higher-order exponential integrals E(x,m,n) are defined in A163931; for information about the asymptotic expansion of E(x,m=1,n) see A130534. The asymptotic expansions for n = 1, 2, 3, 4, ..., lead to the right hand columns of the triangle given above. - Johannes W. Meijer, Oct 16 2009
The number of injective functions from a set of size k to a set of size n. - Dennis P. Walsh, Feb 10 2011
The number of functions f from {1,2,...,k} to {1,2,...,n} that satisfy f(x) >= x for all x in {1,2,...,k}. - Dennis P. Walsh, Apr 20 2011
T(n,k) = A181511(n,k) for k=1..n-1. - Reinhard Zumkeller, Nov 18 2012
The e.g.f.s enumerating the faces of the permutohedra / permutahedra, Perm(s,t;x) = [e^(sx)-1]/[s-t(e^(sx)-1)], (cf. A090582 and A019538) and the stellahedra / stellohedra, St(s,t;x) = [s e^((s+t)x)]/[s-t(e^(sx)-1)], (cf. A248727) given in Toric Topology satisfy exp[u*d/dt] St(s,t;x) = St(s,u+t;x) = [e^(ux)/(1-u*Perm(s,t;x))]*St(s,t;x), where e^(ux)/(1-uy) is a bivariate e.g.f. for the row polynomials of this entry and A094587. Equivalently, d/dt St = (x+Perm)*St and d/dt Perm = Perm^2, or d/dt log(St) = x + Perm and d/dt log(Perm) = Perm. - Tom Copeland, Nov 14 2016
T(n, k)/n! are the coefficients of the n-th exponential Taylor polynomial, or truncated exponentials, which was proved to be irreducible by Schur. See Coleman link. - Michel Marcus, Feb 24 2020
Given a generic choice of k+2 residues, T(n, k) is the number of meromorphic differentials on the Riemann sphere having a zero of order n and these prescribed residues at its k+2 poles. - Quentin Gendron, Jan 16 2025

			Triangle begins:
  1;
  1,  1;
  1,  2,  2;
  1,  3,  6,   6;
  1,  4, 12,  24,   24;
  1,  5, 20,  60,  120,   120;
  1,  6, 30, 120,  360,   720,    720;
  1,  7, 42, 210,  840,  2520,   5040,   5040;
  1,  8, 56, 336, 1680,  6720,  20160,  40320,   40320;
  1,  9, 72, 504, 3024, 15120,  60480, 181440,  362880,  362880;
  1, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800;
  ...
For example, T(4,2)=12 since there are 12 injective functions f:{1,2}->{1,2,3,4}. There are 4 choices for f(1) and then, since f is injective, 3 remaining choices for f(2), giving us 12 ways to construct an injective function. - _Dennis P. Walsh_, Feb 10 2011
For example, T(5,3)=60 since there are 60 functions f:{1,2,3}->{1,2,3,4,5} with f(x) >= x. There are 5 choices for f(1), 4 choices for f(2), and 3 choices for f(3), giving us 60 ways to construct such a function. - _Dennis P. Walsh_, Apr 30 2011

CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 176; 31st ed., p. 215, Section 3.3.11.1.
Maple V Reference Manual, p. 490, numbperm(n,k).

T. D. Noe, Rows n = 0..100 of triangle, flattened
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013-2014.
V. Buchstaber and T. Panov Toric Topology, arXiv:1210.2368v3 [math.AT], 2014.
R. F. Coleman, On the Galois groups of the exponential Taylor polynomials, L’Enseignement Mathématique 33, 183-189, (1987).
Quentin Gendron and Guillaume Tahar, Isoresidual fibration and resonance arrangements, Lett. Math. Phys. 112, No. 2, Paper No. 33, 36 p. (2022).
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients for k rooks on the 1xn board, all heights 1.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963) 31-41.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Yuriy Shablya and Dmitry Kruchinin, Euler-Catalan's Number Triangle and its Application, Proceedings Book of Micropam (The Mediterranean International Conference of Pure & Applied Mathematics and Related Areas 2018), 212-215.
Dennis Walsh, Notes on no-less functions
Eric Weisstein's World of Mathematics, Falling Factorial
Index entries for "core" sequences

Row sums give A000522.
Cf. A001497, A001498, A136572.
T(n,0)=A000012, T(n,1)=A000027, T(n+1,2)=A002378, T(n,3)=A007531, T(n,4)=A052762, and T(n,n)=A000142.
Cf. A019538, A090582, A094587, A248727.

a008279 n k = a008279_tabl !! n !! k
a008279_row n = a008279_tabl !! n
a008279_tabl = iterate f [1] where
   f xs = zipWith (+) ([0] ++ zipWith (*) xs [1..]) (xs ++ [0])
-- Reinhard Zumkeller, Dec 15 2013, Nov 18 2012

/* As triangle */ [[Factorial(n)/Factorial(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 11 2015

with(combstruct): for n from 0 to 10 do seq(count(Permutation(n),size=m), m = 0 .. n) od; # Zerinvary Lajos, Dec 16 2007
seq(seq(n!/(n-k)!,k=0..n),n=0..10); # Dennis P. Walsh, Apr 20 2011
seq(print(seq(pochhammer(n-k+1,k),k=0..n)),n=0..6); # Peter Luschny, Mar 26 2015

Table[CoefficientList[Series[(1 + x)^m, {x, 0, 20}], x]* Table[n!, {n, 0, m}], {m, 0, 10}] // Grid (* Geoffrey Critzer, Mar 16 2010 *)
Table[ Pochhammer[n - k + 1, k], {n, 0, 9}, {k, 0, n}] // Flatten (* or *)
Table[ FactorialPower[n, k], {n, 0, 9}, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 18 2013, updated Jan 28 2016 *)

{T(n, k) = if( k<0 || k>n, 0, n!/(n-k)!)}; /* Michael Somos, Nov 14 2002 */

{T(n, k) = my(A, p); if( k<0 || k>n, 0, if( n==0, 1, A = matrix(n, n, i, j, x + (i==j)); polcoeff( sum(i=1, n!, if( p = numtoperm(n, i), prod(j=1, n, A[j, p[j]]))), k)))}; /* Michael Somos, Mar 05 2004 */

from math import factorial, isqrt, comb
def A008279(n): return factorial(a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))//factorial(a-n+comb(a+1,2)) # Chai Wah Wu, Nov 13 2024

for n in range(8): [falling_factorial(n,k) for k in (0..n)] # Peter Luschny, Mar 26 2015

E.g.f.: Sum T(n,k) x^n/n! y^k = exp(x)/(1-x*y). - Vladeta Jovovic, Aug 19 2002
Equals A007318 * A136572. - Gary W. Adamson, Jan 07 2008
T(n, k) = n*T(n-1, k-1) = k*T(n-1, k-1)+T(n-1, k) = n*T(n-1, k)/(n-k) = (n-k+1)*T(n, k-1). - Henry Bottomley, Mar 29 2001
T(n, k) = n!/(n-k)! if n >= k >= 0, otherwise 0.
G.f. for k-th column k!*x^k/(1-x)^(k+1), k >= 0.
E.g.f. for n-th row (1+x)^n, n >= 0.
Sum T(n, k)x^k = permanent of n X n matrix a_ij = (x+1 if i=j, x otherwise). - Michael Somos, Mar 05 2004
Ramanujan psi_1(k, x) polynomials evaluated at n+1. - Ralf Stephan, Apr 16 2004
E.g.f.: Sum T(n,k) x^n/n! y^k/k! = e^{x+xy}. - Franklin T. Adams-Watters, Feb 07 2006
The triangle is the binomial transform of an infinite matrix with (1, 1, 2, 6, 24, ...) in the main diagonal and the rest zeros. - Gary W. Adamson, Nov 20 2006
G.f.: 1/(1-x-xy/(1-xy/(1-x-2xy/(1-2xy/(1-x-3xy/(1-3xy/(1-x-4xy/(1-4xy/(1-... (continued fraction). - Paul Barry, Feb 11 2009
T(n,k) = Sum_{j=0..k} binomial(k,j)*T(x,j)*T(y,k-j) for x+y = n. - Dennis P. Walsh, Feb 10 2011
From Dennis P. Walsh, Apr 20 2011: (Start)
E.g.f (with k fixed): x^k*exp(x).
G.f. (with k fixed): k!*x^k/(1-x)^(k+1). (End)
For n >= 2 and m >= 2, Sum_{k=0..m-2} S2(n, k+2)*T(m-2, k) = Sum_{p=0..n-2} m^p. S2(n,k) are the Stirling numbers of the second kind A008277. - Tony Foster III, Jul 23 2019

A133314 Coefficients of list partition transform: reciprocal of an exponential generating function (e.g.f.).

Original entry on oeis.org

1, -1, -1, 2, -1, 6, -6, -1, 8, 6, -36, 24, -1, 10, 20, -60, -90, 240, -120, -1, 12, 30, -90, 20, -360, 480, -90, 1080, -1800, 720, -1, 14, 42, -126, 70, -630, 840, -420, -630, 5040, -4200, 2520, -12600, 15120, -5040, -1, 16, 56, -168, 112, -1008, 1344, 70

Offset: 0

Tom Copeland, Oct 18 2007, Oct 29 2007, Nov 16 2007

The list partition transform of a sequence a(n) for which a(0)=1 is illustrated by:
b_0 = 1
b_1 = -a_1
b_2 = -a_2 + 2 a_1^2
b_3 = -a_3 + 6 a_2 a_1 - 6 a_1^3
b_4 = -a_4 + 8 a_3 a_1 + 6 a_2^2 - 36 a_2 a_1^2 + 24 a_1^4
... .
The unsigned coefficients are A049019 with a leading 1. The sign is dependent on the partition as evident from inspection (replace a_n's by -1).
Expressed umbrally, i.e., with the umbral operation (a.)^n := a_n,
  exp(a.x) exp(b.x) = exp[(a.+b.)x] = 1; i.e., (a.+b.)^n = 1 for n=0 and 0 for all other values of n.
Expressed recursively,
b_0 = 1, b_n = -Sum_{j=1..n} binomial(n,j) a_j b_{n-j}; which is conditionally self-inverse, i.e., the roles of a_k and b_k may be reversed with a_0 = b_0 = 1.
Expressed in matrix form, b_n form the first column of B = matrix inverse of A .
A = Pascal matrix diagonally multiplied by a_n, i.e., A_{n,k} = binomial(n,k)* a_{n-k}.
Some examples of reciprocal pairs of sequences under these operations are:
1) A084358 and -A000262 with the first term set to 1.
2) (1,-1,0,0,...) and (0!,1!,2!,3!,...) with the unsigned associated matrices A128229 and A094587.
3) (1,-1,-1,-1,...) and A000670.
4) A000110 and A000587.
5) (1,-2,-2,0,0,0,...) and (0! c_1,1! c_2,2! c_3,3! c_4,...) where c_n = A000129(n) with the associated matrices A110327 and A110330.
6) (1,-2,2,0,0,0,...) and (1!,2!,3!,4!,...).
7) Sequences of rising and signed lowering factorials form reciprocal pairs where a_n = (-1)^n m!/(m-n)! and b_n = (m-1+n)!/(m-1)! for m=0,1,2,... .
Denote the action of the list partition transform on the sequence a. or an invertible matrix M by LPT(a.) = b. or LPT(M)= M^(-1).
If the matrix equation M = exp(T) also holds, then exp[a.*T]*exp[b.*T] = exp[(a.+b.)*T] = I, the identity matrix, because (a.+b.)^n = delta_n, the Kronecker delta with delta_n = 1 and delta_n = 0 otherwise, i.e., (0)^n = delta_n.
Therefore, [exp(a.*T)]^(-1) = exp[b.*T] = exp[LPT(a.)*T] = LPT[exp(a.*T)].
The fundamental Pascal (A007318), unsigned Lah (A105278) and associated Laguerre matrices can be generated by exponentiation of special infinitesimal matrices (see A132440, A132710 and A132681) such that finding LPT(a.) amounts to multiplying the k'th diagonal of the fundamental matrices by a_k for every diagonal followed by matrix inversion and then extraction of the b_n factors from the first column (simplest for the Pascal formulas above).
Conversely, the inverses of matrices formed by diagonally multiplying the three fundamental matrices by a_k are given by diagonally multiplying the fundamental matrices by b_k.
If LPT(M) is defined differently as application of the top formula to a_n = M^n, then b_n = (-M)^n and the formalism could even be applied to more general sequences of matrices M., providing the reciprocal of exp[t*M.].
The group of fundamental lower triangular matrices M = exp(T) such that LPT[exp(a.*T)] = exp[LPT(a.)*T] = [exp[a.*T]]^(-1) are obtained by infinitesimal generator matrices of the form T =
  0;
  t(0), 0;
  0, t(1), 0;
  0, 0, t(2), 0;
  0, 0, 0, t(3), 0;
  ... .
T^m has trivially vanishing terms except along the m'th subdiagonal, which is a sequence of generalized factorials:
[ t(0)*t(1)...t(m-2)*t(m-1), t(1)*t(2)...t(m-1)*t(m), t(2)*t(3)...t(m)*t(m+1), ... ].
Therefore the principal submatrices of T (given by setting t(j) = 0 for j > n-1) are nilpotent with at least [Tsub_n]^(n+1) = 0.
The general group of matrices GM[a.] = exp[a.*T] can also be obtained through diagonal multiplication of M = exp(T) by the sequence a_n, as in the Pascal matrix example above and their inverses by diagonal multiplication by b. = LPT(a.).
Weighted-mappings interpretation for the top partition equation:
Given n pre-nodes (Pre) and k post-nodes (Post), each Pre is connected to only one Post and each Post has at least one Pre connected to it (surjections or onto functions/maps). Weight each Post by -a_m where m is the number of connections to the Post.
Weight each map by the product of the Post weights and multiply by the number of maps that share the same connectivity. Sum over the possible mappings for n Pre. The result is b_n.
E.g., b_3 = [ 3 Pre to 1 Post ] + [ 3 Pre to 2 Post ] + [ 3 Pre to 3 Post ]
= [1 map with 1 Post with 3 connections] + [ 6 maps with 1 Post with 2 connections and 1 Post with 1 connection] + [6 maps with 3 Post with 1 connection each]
= -a_3 + 6 * [-a_2*(-a_1)] + 6 * [-a_1*(-a_1)*(-a_1)].
See A263633 for the complementary formulation for the reciprocal of o.g.f.s rather than e.g.f.s and computations of these partition polynomials as Gram determinants. - Tom Copeland, Dec 04 2016
The coefficients of the partition polynomials enumerate the faces of the convex, bounded polytopes called permutohedra, and the absolute value of the sum of the coefficients gives the Euler characteristic of unity for each polytope; i.e., the absolute value of the sum of each row of the array is unity. In addition, the signs of the faces alternate with dimension, and the coefficients of faces with the same dimension for each polytope have the same sign. - Tom Copeland, Nov 13 2019
With the fundamental matrix chosen to be the lower triangular Pascal matrix M, the matrix MA whose n-th diagonals are multiplied by a_n (i.e., MA_{i,j} = PM_{i,j} * a_{i-j}) gives a matrix representation of the e.g.f. associated to the Appell polynomial sequence defined by e^{a.t}e^{xt}= e^{(a.+x)t} = e^{A.(x)t} where umbrally (A.(x))^n = A_n(x) = (a. + x)^n = sum_{k=0..n} binomial(n,k) a_k x^{n-k} are the associated Appell polynomials. Left multiplication of the column vector (1,x,x^2,..) by MA gives the Appell polynomial sequence, and multiplication of the two e.g.f.s e^{a.t} and e^{b.t} corresponds to multiplication of their respective matrix representations MA and MB. Forming the reciprocal of an e.g.f. corresponds to taking the matrix inverse of its matrix representation as noted above. A263634 gives an associated modified Pascal matrix representation of the raising operator for the Appell sequence. - Tom Copeland, Nov 13 2019
The diagonal of MA consists of all ones. Let MAN be the truncated square submatrix of MA containing the coefficients of the first N Appell polynomials A_k=(a.+x)^k = Sum(j=0 to k) MAN(k,j) x^j. Then by the Cayley-Hamilton theorem (I-MAN)^N = 0; therefore, MAN^(-1) = Sum(k=1 to N) binomial(N,k) (-MAN)^{k-1} = MBN, the inverse of MAN, containing the coefficients of the first N rows of the Appell polynomials B_k(x) = (b. + x)^k = Sum(j=0 to k) MBN(k,j) x^j, which are the umbral compositional inverses of the Appell row polynomials A_k(x) of MAN; that is, A_k(B.(x)) = x^k = B_k(A.(x)), where, e.g., (A.(x))^k = A_k(x). - Tom Copeland, May 13 2020
The use of the term 'list partition transform' resulted from one of my first uses of these partition polynomials in relating A000262 to A084358 with their simple e.g.f.s. Other appropriate names would be the permutohedra polynomials since they are refined Euler characteristics of the permutohedra or the reciprocal polynomials since they give the multiplicative inverses of e.g.f.s with a constant of 1. - Tom Copeland, Oct 09 2022

			Table starts:
[0] [ 1]
[1] [-1]
[2] [-1,  2]
[3] [-1,  6, -6]
[4] [-1,  8,  6, -36,  24]
[5] [-1, 10, 20, -60, -90,  240, -120]
[6] [-1, 12, 30, -90,  20, -360,  480, -90, 1080, -1800, 720]

Vincenzo Librandi, Table of n, a(n) for n = 0..250
M. Aguiar, Math 7410: Lie Combinatorics and Hyperplane Arrangements, notes made by D. Mehrle for course taught by Aguiar at Cornell, 2016, p. 49.
M. Aguiar and F. Ardila, The algebraic and combinatorial structure of generalized permutahedra [sic], MSRI Summer School July 19, 2017.
M. Aguiar and F. Ardila, Hopf monoids and generalized permutahedra [sic], arXiv:1709.07504 [math.CO], p. 5, 2017.
N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, Scattering forms and the positive geometry of kinematics, color, and the worldsheet , arXiv:1711.09102 [hep-th], 2017.
J. Bergström and S. Minabe, On the cohomology of the Losev-Manin moduli space, arXiv:1108.0338 [math.AG], (cf. p. 1), 2013.
Z. Bern, J.Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban , The Duality Between Color and Kinematics and its Applications, arXiv preprint arXiv:1909.01358 [hep-th], 2019.
L. Berry, S. Forcey, M. Ronco, and P. Showers, Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra, arXiv:1608.08546 [math.CO], 2018.
Tom Copeland, Bijective mapping between face polytopes of permutohedra and partitions of integers, Math StackExchange question, 2016.
Tom Copeland, Compilation of OEIS Partition Polynomials A133314, A134685, A145271, A356144, and A356145, 2022.
Tom Copeland, In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019.
Karl-Dieter Crisman, The Borda Count, the Kemeny Rule, and the permutahedron [sic], preprint, 2014.
Karl-Dieter Crisman, The Borda Count, the Kemeny Rule, and the permutahedron [sic], in: Karl-Dieter Crisman and Michael A. Jones (eds.), The Mathematics of Decisions, Elections, and Games, Contemporary Mathematics, AMS, Vol. 624, 2014, pp. 101-134.
R. Da Rosa, D. Jensen, and D. Ranganathan, Toric graph associahedra and compactifications of M_(0,n), arXiv:1411.0537 [math.AG], 2015.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
S. Forcey, The Hedra Zoo
S. Franco and A. Hasan, Graded Quivers, Generalized Dimer Models and Toric Geometry , arXiv preprint arXiv:1904.07954 [hep-th], 2019.
M. Futaki and K. Ueda, Tropical coamoeba and torus-equivariant homological mirror symmetry for the projective space , arXiv preprint arXiv:1001.4858 [math.SG], 2010-2014.
X. Gao, S. He, Y. Zhan, Labelled tree graphs, Feynman diagrams and disk integrals , arXiv:1708.08701 [hep-th], 2017.
A. Hasan, Physics and Mathematics of Graded Quivers, dissertation, Graduate Center, City University of New York, 2019.
D. Karp, D. Ranganathan, P. Riggins, and U. Whitcher, Toric Symmetry of CP^3, arXiv:1109.5157 [math.AG], 2011.
R. Kaufmann and Y. Zhang, Permutohedral structures on E2-operads, arXiv preprint arXiv:1602.08247 [math.AT], 2016.
M. Lin, Graph Cohomology, 2016.
J. Loday, The Multiple Facets of the Associahedron
MathOverflow, Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions, an MO question posed by T. Copeland, 2017.
W. Norledge and A. Ocneanu, Hopf monoids, permutohedral cones, and generalized retarded functions, arXiv preprint arXiv:1911.11736 [math.CO], 2020.
P. Olver, The canonical contact form, p. 9.
V. Pilaud, The Associahedron and its Friends, presentation for Séminaire Lotharingien de Combinatoire, April 4 - 6, 2016.
J . Pitman and R. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, arxiv: 9908029 [math.CO], 1999.
A. Postnikov, Positive Grassmannian and Polyhedral Subdivisions, arXiv:1806.05307 [math.CO], (cf. p. 17), 2018.
D. Ranganathan, Gromov-Witten Theory of Blowups of Toric Threefolds, senior thesis, Harvey Mudd College, 2012.
D. Ranganathan, Gromov-Witten Theory of Blowups of Toric Threefolds (poster), poster for senior thesis, Harvey Mudd College, 2012.
E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen vol. 2, 317-365, 1870, (bottom of p. 342, see Stewart below for a translation).
G. Stewart, On infinitely many algorithms for solving equations, 1993, (translation into English of the Schröder paper above, see top of p. 31 for differently normalized partition polynomials of this entry).

Row lengths in A000041.
Cf. A000110, A000129, A000262, A000587, A000670, A001286, A007318, A019538, A049019, A084358, A094587, A105278, A110327, A110330, A128229, A132440, A132681, A132710, A263633.
Cf. A134685, A145271, A356144, A356145.

b[0] = 1; b[n_] := b[n] = -Sum[Binomial[n, j]*a[j]*b[n-j], {j, 1, n}];
row[0] = {1}; row[n_] := Coefficient[b[n], #]& /@ (Times @@ (a /@ #)&) /@ IntegerPartitions[n];
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Apr 23 2014 *)

def A133314_row(n): return [(-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in Partitions(n)]
for n in (0..10): print(A133314_row(n)) # Peter Luschny, Sep 18 2015

b_{n-1} = (1/n)(d/da(1))p_n[a_1, a_2, ..., a_n] where p_n are the row partition polynomials of the cumulant generator A127671. - Tom Copeland, Oct 13 2012
(E.g.f. of matrix B) = (e.g.f. of b)·exp(xt) = exp(b.t)·exp(xt) = exp(xt)/exp(a.t) = (e.g.f. of A^(-1)) and (e.g.f. of matrix A) = exp(a.t)·exp(xt) = exp(xt)/exp(b.t) = (e.g.f. of B^(-1)), where the umbral evaluation of exp(b.t) = Sum{n >= 0} (b.t)^n / n! = Sum_{n >= 0} b_n t^n / n! is understood in the denominator. These e.g.f.s define Appell sequences of polynomials. - Tom Copeland, Mar 22 2014
Sum of the n-th row is (-1)^n. - Peter Luschny, Sep 18 2015
The unsigned coefficients for the partitions a_2*a_1^n for n >= 0 are the Lah numbers A001286. - Tom Copeland, Aug 06 2016
G.f.: 1 / (1 + Sum_{n > 0} a_n x^n/n!) = 1 / exp(a.x). - Tom Copeland, Oct 18 2016
Let a_1 = 1 + x + B_1 = x + 1/2 and a_n = B_n = (B.)^n, where B_n are the Bernoulli numbers defined by e^(B.t) = t / (e^t-1), then t / e^(a.t) = t / [(x + 1) * t + exp(B.t)] =  (e^t - 1) /[ 1 + (x + 1) (e^t - 1)] = exp(p.(x)t), where (p.(x))^n = p_n(x) are the shifted signed polynomials of A019538: p_0(x) = 0, p_1(x) = 1, p_2(x) = -(1 + 2 x), p_3(x) = 1 + 6 x + 6 x^2, ... , p_n(x) = n * b_{n-1}. - Tom Copeland, Oct 18 2016
With a_n = 1/(n+1), b_n = B_n, the Bernoulli numbers. - Tom Copeland, Nov 08 2016
Indeterminate substitutions as illustrated in A356145 lead to [E] = [L][P] = [P][E]^(-1)[P] = [P][RT] and [E]^(-1) = [P][L] = [P][E][P] = [RT][P], where [E] contains the refined Eulerian partition polynomials of A145271; [E]^(-1), A356145, the inverse set to [E]; [P], the permutohedra polynomials of this entry; [L], the classic Lagrange inversion polynomials of A134685; and [RT], the reciprocal tangent polynomials of A356144. Since [L]^2 = [P]^2 = [RT]^2 = [I], the substitutional identity, [L] = [E][P] = [P][E]^(-1) = [RT][P], [RT] = [E]^(-1)[P] = [P][L][P] = [P][E], and [P] = [L][E] = [E][RT] = [E]^(-1)[L] = [RT][E]^(-1). - Tom Copeland, Oct 05 2022

More terms from Jean-François Alcover, Apr 23 2014

A036039 Irregular triangle of multinomial coefficients of integer partitions read by rows (in Abramowitz and Stegun ordering) giving the coefficients of the cycle index polynomials for the symmetric groups S_n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 8, 3, 6, 1, 24, 30, 20, 20, 15, 10, 1, 120, 144, 90, 40, 90, 120, 15, 40, 45, 15, 1, 720, 840, 504, 420, 504, 630, 280, 210, 210, 420, 105, 70, 105, 21, 1, 5040, 5760, 3360, 2688, 1260, 3360, 4032, 3360, 1260, 1120, 1344, 2520, 1120, 1680, 105, 420, 1120, 420, 112, 210, 28, 1

Offset: 1

N. J. A. Sloane

The sequence of row lengths is A000041(n), n >= 1 (partition numbers).
Number of permutations whose cycle structure is the given partition. Row sums are factorials (A000142). - Franklin T. Adams-Watters, Jan 12 2006
A relation between partition polynomials formed from these "refined" Stirling numbers of the first kind and umbral operator trees and Lagrange inversion is presented in the link "Lagrange a la Lah".
These cycle index polynomials for the symmetric group S_n are also related to a raising operator / infinitesimal generator for fractional integro-derivatives, involving the digamma function and the Riemann zeta function values at positive integers, and to the characteristic polynomial for the adjacency matrix of complete n-graphs A055137 (cf. MathOverflow link). - Tom Copeland, Nov 03 2012
In the Lang link, replace all x(n) by t to obtain A132393. Furthermore replace x(1) by t and all other x(n) by 1 to obtain A008290. See A274760. - Tom Copeland, Nov 06 2012, Oct 29 2015 - corrected by Johannes W. Meijer, Jul 28 2016
The umbral compositional inverses of these polynomials are formed by negating the indeterminates x(n) for n>1, i.e., P(n,P(.,x(1),-x(2),-x(3),...),x(2),x(3),...) = x(1)^n (cf. A130561 for an example of umbral compositional inversion). The polynomials are an Appell sequence in x(1), i.e., dP(n,x(1))/dx(1) = n P(n-1, x(1)) and (P(.,x)+y)^n=P(n,x+y) umbrally, with P(0,x(1))=1. - Tom Copeland, Nov 14 2014
Regarded as the coefficients of the partition polynomials listed by Lang, a signed version of these polynomials IF(n,b1,b2,...,bn) (n! times polynomial on page 184 of Airault and Bouali) provides an inversion of the Faber polynomials F(n,b1,b2,...,bn) (page 52 of Bouali, A263916, and A115131). For example, F(3, IF(1,b1), IF(2,b1,b2)/2!, IF(3,b1,b2,b3)/3!) = b3 and IF(3, F(1,b1), F(2,b1,b2), F(3,b1,b2,b3))/3! = b3 with F(1,b1) = -b1. (Compare with A263634.) - Tom Copeland, Oct 28 2015; Sep 09 2016
The e.g.f. for the row partition polynomials is Sum_{n>=0} P_n(b_1,...,b_n) x^n/n! = exp[Sum_{n>=1} b_n x^n/n], or, exp[P.(b_1,...,b_n)x] = exp[-], expressed umbrally with <"power series"> denoting umbral evaluation (b.)^n = b_n within the power series. This e.g.f. is central to the paper by Maxim and Schuermannn on characteristic classes (cf. Friedrich and McKay also). - Tom Copeland, Nov 11 2015
The elementary Schur polynomials are given by S(n,x(1),x(2),...,x(n)) = P(n,x(1), 2*x(2),...,n*x(n)) / n!. See p. 12 of Carrell. - Tom Copeland, Feb 06 2016
These partition polynomials are also related to the Casimir invariants associated to quantum density states on p. 3 of Boya and Dixit and pp. 5 and 6 of Byrd and Khaneja. - Tom Copeland, Jul 24 2017
With the indeterminates (x_1,x_2,x_3,...) = (t,-c_2*t,-c_3*t,...) with c_n >0, umbrally P(n,a.) = P(n,t)|{t^n = a_n} = 0 and P(j,a.)P(k,a.) = P(j,t)P(k,t)|{t^n =a_n} = d_{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)], where a_n are the inversion partition polynomials for calculating f(x) from the coefficients of the series expansion of f^{-1}(x) given in A133932. - Tom Copeland, Feb 09 2018
For relation to the Witt symmetric functions, as well as the basic power, elementary, and complete symmetric functions, see the Borger link p. 295. For relations to diverse zeta functions, determinants, and paths on graphs, see the MathOverflow question Cycling Through the Zeta Garden. - Tom Copeland, Mar 25 2018
Chmutov et al. identify the partition polynomials of this entry with the one-part Schur polynomials and assert that any linear combination with constant coefficients of these polynomials is a tau function for the KP hierarchy. - Tom Copeland, Apr 05 2018
With the indeterminates in the partition polynomials assigned as generalized harmonic numbers, i.e., as partial sums of the Dirichlet series for the Riemann zeta function, zeta(n), for integer n > 1, sums of simple normalizations of these polynomials give either unity or simple sums of consecutive zeta(n) (cf. Hoffman). Other identities involving these polynomials can be found in the Choi reference in Hoffman's paper. - Tom Copeland, Oct 05 2019
On p. 39 of Ma Luo's thesis is the e.g.f. of rational functions r_n obtained through the (umbral) formula 1/(1-r.T) = exp[log(1+P.T)], a differently signed e.g.f. of this entry, where (P.)^n = P_n are Eisenstein elliptic functions. P. 38 gives the example of 4! * r_4 as the signed 4th row partition polynomial of this entry. This series is equated through a simple proportionality factor to the Zagier Jacobi form on p. 25. Recurrence relations for the P_n are given on p. 24 involving the normalized k-weight Eisenstein series G_k introduced on p. 23 and related to the Bernoulli numbers. - Tom Copeland, Oct 16 2019
The Chern characteristic classes or forms of complex vector bundles and the characteristic polynomials of curvature forms for a smooth manifold can be expressed in terms of this entry's partition polynomials with the associated traces, or power sum polynomials, as the indeterminates. The Chern character is the e.g.f. of these traces and so its coefficients are given by the Faber polynomials with this entry's partition polynomials as the indeterminates. See the Mathoverflow question "A canonical reference for Chern characteristic classes". - Tom Copeland, Nov 04 2019
For an application to the physics of charged fermions in an external field, see Figueroa et al. - Tom Copeland, Dec 05 2019
Konopelchenko, in Proposition 5.2, p. 19, defines an operator P_k that is a differently signed operator version of the partition polynomials of this entry divided by a factorial. These operators give rise to bilinear Hirota equations for the KP hierarchy. These partition polynomials are also presented in Hopf algebras of symmetric functions by Cartier. - Tom Copeland, Dec 18 2019
For relationship of these partition polynomials to calculations of Pontryagin classes and the Riemann xi function, see A231846. - Tom Copeland, May 27 2020
Luest and Skliros summarize on p. 298 many of the properties of the cycle index polynomials given here; and Bianchi and Firrotta, a few on p. 6. - Tom Copeland, Oct 15 2020
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)

			The partition array T(n, k) begins (see the W. Lang link for rows 1..10):
  n\k   1    2    3    4    5    6    7    8    9   10   11  12   13  14 15 ...
  1:    1
  2:    1    1
  3:    2    3    1
  4:    6    8    3    6    1
  5:   24   30   20   20   15   10    1
  6:  120  144   90   40   90  120   15   40   45   15    1
  7:  720  840  504  420  504  630  280  210  210  420  105  70  105  21  1
... reformatted by _Wolfdieter Lang_, May 25 2019

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_2".

David W. Wilson, Table of n, a(n) for n = 1..11731 (rows 1 through 26).
Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.
H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, Pages 179-222.
M. Bianchi and M. Firrotta, DDF operators, open string coherent states and their scattering amplitudes, arXiv:1902.07016 [hep-th], 2019.
J. Borger, Witt vectors, semirings, and total positivity, arXiv:1310.3013 [math.CO], 2015.
A. Bouali, Faber polynomials Cayley-Hamilton equation and Newton symmetric functions, Bulletin des Sciences Mathématiques, Volume 130, Issue 1, Jan-Feb 2006, Pages 49-70.
L. Boya and K. Dixit, Geometry of density states, arXiv:808.1930 [quant-phy], 2017.
M. Byrd and N. Khaneja, Characterization of the positivity of the density matrix in terms of the coherence vector representation , arXiv:quant-phy/0302024, 2003.
S. Carrell, Combinatorics of the KP Hierarchy, Thesis, University of Waterloo, Ontario, Canada, 2009.
P. Cartier, A primer of Hopf algebras, preprint, Institut des Hautes Etudes Scientifiques, France, 2006, pp. 56 and 57.
S. Chmutov, M. Kazarian, and S. Lando, Polynomial graph invariants and the KP hierarchy , arXiv:1803.09800 [math.CO], p. 16-17, 2018.
Tom Copeland, Lagrange a la Lah, 2011; Riemann zeta function at positive integers and an Appell sequence of polynomials, 2012; The creation / raising operators for Appell sequences, 2015; Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015; Addendum to Elliptic Lie Triad, 2015; Formal group laws and binomial Sheffer polynomials, 2018; Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials, 2020.
Mark Dominus, Cycle classes of permutations [From _Wouter Meeussen_, Jun 26 2009]
G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015.
H. Figueroa and J. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (pp. 40-41).
R. Friedrich and J. McKay, Formal groups, Witt vectors and free probability, arXiv:1204.6522 [math.OA], 2012.
M. Hoffman, Harmonic-number summation identities, symmetric functions, and multiple zeta values, arXiv:1602.03198 [math.NT], p. 2-3, 2016.
B. Konopelchenko, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
Wolfdieter Lang, First ten rows and polynomials.
D. Luest and D. Skliros, Handle Operators in String Theory, arXiv:1912.01055 [hep-th], 2019.
Ma Luo, Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves, Ph.D. dissertation, Math Dept., Duke Univ., 2018.
Ma Luo, Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves, arXiv:1804.06977 [math.NT], 2019.
MathOverflow, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants, a question posed by Tom Copeland, 2012.
MathOverflow, Canonical reference for Chern characteristic classes, a question posed by Tom Copeland, 2019.
L. Maxim and J. Schuermann, Equivariant characteristic classes of external and symmetric products of varieties, arXiv:1508.04356 [math.AG], 2015.
R. Szabo, N=2 gauge theories, instanton moduli spaces and geometric representation theory, arXiv:1507.00685 [hep-th], 2015.
Andrei Vieru, Analytic renormalization of multiple zeta functions. Geometry and combinatorics of generalized Euler reflection formula for MZV, arXiv preprint arXiv:1601.04703 [math.NT], 2016.

Cf. A036036, A036037, A036038, A036040.
Cf. other versions based on different partition orderings: A102189 (rows reversed), A181897, A319192.
Cf. A094587, A115131, A130561, A263634, A263916.
Cf. A133932.
Cf. A231846.
Cf. A127671.

nmax:=7: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while sA036039(n, m) := n!/ (mul((t)^q(t)*q(t)!, t=1..n)); od: od: seq(seq(A036039(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016
# 2nd program:
A036039 := proc(n,k)
    local a,prts,e,ai ;
    a := n! ;
    # ASPrts is implemented in A119441
    prts := ASPrts(n)[k] ;
    ai := 1;
    for e from 1 to nops(prts) do
        if e>1 then
            if op(e,prts) = op(e-1,prts) then
                ai := ai+1 ;
            else
                ai := 1;
            end if;
        end if;
        a := a/(op(e,prts)*ai) ;
    end do:
    a ;
end proc:
seq(seq(A036039(n,k),k=1..combinat[numbpart](n)),n=1..15) ; # R. J. Mathar, Dec 18 2016

aspartitions[n_]:=Reverse/@Sort[Sort/@IntegerPartitions[n]];(* Abramowitz & Stegun ordering *);
ascycleclasses[n_Integer]:=n!/(Times@@ #)&/@((#!
Range[n]^#)&/@Function[par,Count[par,# ]&/@Range[n]]/@aspartitions[n])
(* The function "ascycleclasses" is then identical with A&S multinomial M2. *)
Table[ascycleclasses[n], {n, 1, 8}] // Flatten
(* Wouter Meeussen, Jun 26 2009, Jun 27 2009 *)

def PartAS(n):
    P = []
    for k in (1..n):
        Q = [p.to_list() for p in Partitions(n, length=k)]
        for q in Q: q.reverse()
        P = P + sorted(Q)
    return P
def A036039_row(n):
    fn, C = factorial(n), []
    for q in PartAS(n):
        q.reverse()
        p = Partition(q)
        fp = 1; pf = 1
        for a, c in p.to_exp_dict().items():
            fp *= factorial(c)
            pf *= factorial(a)**c
        co = fn//(fp*pf)
        C.append(co*prod([factorial(i-1) for i in p]))
    return C
for n in (1..10):
    print(A036039_row(n)) # Peter Luschny, Dec 18 2016

T(n,k) = n!/Product_{j=1..n} j^a(n,k,j)*a(n,k,j)!, with the k-th partition of  n >= 1 in Abromowitz-Stegun order written as Product_{j=1..n} j^a(n,k,j) with nonnegative integers a(n,k,j) satisfying Sum_{j=1..n} j*a(n,k,j) = n, and the number of parts is Sum_{j=1..n} a(n,k,j) =: m(n,k). - Wolfdieter Lang, May 25 2019
Raising and lowering operators are given for the partition polynomials formed from this sequence in the link in "Lagrange a la Lah Part I" on p. 23. - Tom Copeland, Sep 18 2011
From Szabo p. 34, with b_n = q^n / (1-q^n)^2, the partition polynomials give an expansion of the MacMahon function M(q) = Product_{n>=1} 1/(1-q^n)^n = Sum_{n>=0} PL(n) q^n, the generating function for PL(n) = n! P_n(b_1,...,b_n), the number of plane partitions with sum n. - Tom Copeland, Nov 11 2015
From Tom Copeland, Nov 18 2015: (Start)
The partition polynomials of A036040 are obtained by substituting x[n]/(n-1)! for x[n] in the partition polynomials of this entry.
CIP_n(t-F(1,b1),-F(2,b1,b2),...,-F(n,b1,...,bn)) = P_n(b1,...,bn;t), where CIP_n are the partition polynomials of this entry; F(n,...), those of A263916; and P_n, those defined in my formula in A094587, e.g., P_2(b1,b2;t) = 2 b2 + 2 b1 t + t^2.
CIP_n(-F(1,b1),-F(2,b1,b2),...,-F(n,b1,...,bn)) = n! bn. (End)
From the relation to the elementary Schur polynomials given in A130561 and above, the partition polynomials of this array satisfy (d/d(x_m)) P(n,x_1,...,x_n) = (1/m) * (n!/(n-m)!) * P(n-m,x_1,...,x_(n-m)) with P(k,...) = 0 for k<0. - Tom Copeland, Sep 07 2016
Regarded as Appell polynomials in the indeterminate x(1)=u, the partition polynomials of this entry P_n(u) obey d/du P_n(u) = n * P_{n-1}(u), so the abscissas for the zeros of P_n(u) are the same as those of the extrema of P{n+1}(u). In addition, the coefficient of u^{n-1} in P_{n}(u) is zero since these polynomials are related to the characteristic polynomials of matrices with null main diagonals, and, therefore, the trace is zero, further implying the abscissa for any zero is the negative of the sum of the abscissas of the remaining zeros. This assumes all zeros are distinct and real. - Tom Copeland, Nov 10 2019

More terms from David W. Wilson

Title expanded by Tom Copeland, Oct 15 2020

cp's OEIS Frontend

A128229 A natural number transform, inverse of signed A094587.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A131202 A coefficient tree from the list partition transform relating A111884, A084358, A000262, A094587, A128229 and A131758.

Views

Author

Keywords

Comments

Programs

Mathematica

PARI

Formula

Extensions

A126064 Triangle read by rows, obtained by multiplying columns of triangle in A094587 by 1,2,4,8,16,... respectively.

Views

Author

Keywords

Comments

Examples

Links

Programs

Haskell

Maple

Mathematica

Extensions

A375466 Array read by ascending antidiagonals of triangles read by rows: the coefficients of the polynomials n! * m^(n-k) * x^k * A094587(n, k), for m >= 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A000522 Total number of ordered k-tuples (k=0..n) of distinct elements from an n-element set: a(n) = Sum_{k=0..n} n!/k!.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

PARI

PARI

PARI

PARI

Sage

Formula

Extensions

A001710 Order of alternating group A_n, or number of even permutations of n letters.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI