A001563 -id:A001563 - cp's OEIS frontend

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

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

A002720 Number of partial permutations of an n-set; number of n X n binary matrices with at most one 1 in each row and column.

Original entry on oeis.org

1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114, 234662231, 3405357682, 53334454417, 896324308634, 16083557845279, 306827170866106, 6199668952527617, 132240988644215842, 2968971263911288999, 69974827707903049154, 1727194482044146637521, 44552237162692939114282

Offset: 0

N. J. A. Sloane

a(n) is also the total number of increasing subsequences of all permutations of [1..n] (see Lifschitz and Pittel). - N. J. A. Sloane, May 06 2012
a(n) = A000142 + A001563 + A001809 + A001810 + A001811 + A001812 + ... these sequences respectively give the number of increasing subsequences of length i for i=0,1,2,... in all permutations of [1..n]. - Geoffrey Critzer, Jan 17 2013
a(n) is also the number of matchings in the complete bipartite graph K(n,n). - Sharon Sela (sharonsela(AT)hotmail.com), May 19 2002
a(n) is also the number of 12-avoiding signed permutations in B_n (see Simion ref).
a(n) is also the order of the symmetric inverse semigroup (monoid) I_n. - A. Umar, Sep 09 2008
EXP transform of A001048(n) = n! + (n-1)!. - Franklin T. Adams-Watters, Dec 28 2006
From Peter Luschny, Mar 27 2011: (Start)
Let B_{n}(x) = Sum_{j>=0} exp(j!/(j-n)!*x-1)/j!; then a(n) = 2! [x^2] Taylor(B_{n}(x)), where [x^2] denotes the coefficient of x^2 in the Taylor series for B_{n}(x).
a(n) is column 2 of the square array representation of A090210. (End)
a(n) is the Hosoya index of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Jul 09 2011
a(n) is also number of non-attacking placements of k rooks on an n X n board, summed over all k >= 0. - Vaclav Kotesovec, Aug 28 2012
Also the number of vertex covers and independent vertex sets in the n X n rook graph. - Eric W. Weisstein, Jan 04 2013
a(n) is the number of injective functions from subsets of [n] to [n] where [n]={1,2,...,n}. For a subset D of size k, there are n!/(n-k)! injective functions from D to [n]. Summing over all subsets, we obtain a(n) = Sum_{k=0..n} C(n,k)*n!/(n-k)! = Sum_{k=0..n} k!*C(n,k)^2. - Dennis P. Walsh, Nov 16 2015
Also the number of cliques in the n X n rook complement graph. - Eric W. Weisstein, Sep 14 2017
a(n)/n! is the expected value of the n-th term of Ulam's "history-dependent random sequence". See Kac (1989), Eq.(2). - N. J. A. Sloane, Nov 16 2019
a(2*n) is odd and a(2*n+1) is even for all n. More generally, for each positive integer k, a(n+k) == a(n) (mod k) for all n. It follows that for each positive integer k, the sequence obtained by reducing a(n) modulo k is periodic, with period dividing k. Various divisibility properties of the sequence follow from this: for example, a(7*n+2) == 0 (mod 7), a(11*n+4) == 0 (mod 11), a(17*n+3) == 0 (mod 17) and a(19*n+4) == 0 (mod 19). - Peter Bala, Nov 07 2022
Conjecture: a(n)*k is the sum of the largest parts in all integer partitions containing their own first differences with n + 1 parts and least part k. - John Tyler Rascoe, Feb 28 2024

			G.f. = 1 + 2*x + 7*x^2 + 34*x^3 + 209*x^4 + 1546*x^5 + 13327*x^6 + 130922*x^7 + ... - _Michael Somos_, Jul 31 2018

J. M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995). [From A. Umar, Sep 09 2008]
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
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).
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 356.

Alois P. Heinz, Table of n, a(n) for n = 0..443 (first 101 terms from T. D. Noe)
Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Ramified inverse and planar monoids, arXiv:2210.17461 [math.RT], 2022.
A. I. Aptekarev, On linear forms containing the Euler constant, arXiv:0902.1768 [math.NT], 2009.
T. Banica, The algebraic structure of quantum partial isometries, arXiv:1411.0577 [math.OA], 2014-2015.
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.
Teo Banica, Algebraic invariants of truncated Fourier matrices, arXiv preprint arXiv:1401.5023 [math.QA], 2014.
D. Borwein, S. Rankin, and L. Renner, Enumeration of injective partial transformations, Discrete Math. (1989), 73: 291-296. [From A. Umar, Sep 09 2008]
D. Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.
Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024. See p. 14.
Fabio Deelan Cunden, Jakub Czartowski, Giovanni Gramegna, and A. de Oliveira Junior, Relative volume of comparable pairs under semigroup majorization, arXiv:2410.23196 [math-ph], 2024. See pp. 4, 15.
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 598.
Olof Giselsson, The Universal C*-Algebra of the Quantum Matrix Ball and its Irreducible *-Representations, arXiv:1801.10608 [math.QA], 2018.
J. Godbout, Combinatorial Properties of the Mirabolic RSK Algorithm, Thesis presented to The Faculty of the Graduate College of The University of Vermont, May 2013.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 64.
Z. Janelidze, F. van Niekerk, and J. Viljoen, What is the maximal connected partial symmetry index of a connected graph of a given size?, arXiv:2502.00704 [math.CO], 2025. See p. 3.
Mark Kac, A history-dependent random sequence defined by Ulam, Advances in Applied Mathematics 10.3 (1989): 270-277.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 219.
Vaclav Kotesovec, Too many errors around coefficient C1 in asymptotic of sequence A002720, Sep 28 2012. [The bug in program Mathematica was fixed in version 10.2.0.0 / July 2015. _Vaclav Kotesovec_, Jul 25 2015]
V. Lifschitz and P. Pittel, The number of increasing subsequences of the random permutation J. Combin. Theory Ser. A 31 (1981), no. 1, 1--20. MR0626437 (84e:05012)
Mathematica Stack Exchange, Wrong Limit with LaguerreL, May 22 2015
W. D. Munn, The characters of the symmetric inverse semigroup, Proc. Cambridge Philos. Soc. 53 (1957), 13-18. [From A. Umar, Sep 09 2008]
K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp, and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009).
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.
John Riordan, Letter, Apr 28 1976.
John Riordan, Letter to N. J. A. Sloane, Oct 01 1978.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
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)
R. Simion, Combinatorial statistics on type-B analogues of non-crossing partitions and restricted permutations, Electronic J. of Comb. 7 (2000), Art #R9.
A. Umar, Some combinatorial problems in the theory of symmetric ..., Algebra Disc. Math. 9 (2010) 115-126.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
D. P. Walsh, Notes on functions from subsets of {1,2,...,n} into {1,2,...,n}.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Matching.
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Eric Weisstein's World of Mathematics, Rook Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for sequences related to Laguerre polynomials

Main diagonal of A088699. Column of A283500. Row sums of A144084.
Cf. A000110, A020556, A069223, A000712, A001048, A090210, A002793, A073003, A000262.
Column k=1 of A289192.
Cf. A160617, A160618.
Cf. A364673.

[Factorial(n)*Evaluate(LaguerrePolynomial(n), -1): n in [0..25]]; // G. C. Greubel, Aug 11 2022

A002720 := proc(n) exp(-x)*n!*hypergeom([n+1], [1], x); simplify(subs(x=1, %)) end: seq(A002720(n), n=0..25); # Peter Luschny, Mar 30 2011
A002720 := proc(n)
    option remember;
    if n <= 1 then
        n+1 ;
    else
        2*n*procname(n-1)-(n-1)^2*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Mar 09 2017

Table[n! LaguerreL[n, -1], {n, 0, 25}]
Table[(-1)^n*HypergeometricU[-n, 1, -1], {n, 0, 25}] (* Jean-François Alcover, Jul 15 2015 *)
RecurrenceTable[{(n+1)^2 a[n] - 2(n+2) a[n+1] + a[n+2]==0, a[1]==2, a[2]==7}, a, {n, 25}] (* Eric W. Weisstein, Sep 27 2017 *)

a(n) = sum(k=0, n, k!*binomial(n, k)^2 );

a(n) = suminf ( k=0, binomial(n+k,n)/k! ) / ( exp(1)/n! ) /* Gottfried Helms, Nov 25 2006 */

{a(n)=n!^2*polcoeff(exp(x+x*O(x^n))*sum(m=0,n,x^m/m!^2),n)} /* Paul D. Hanna, Nov 18 2011 */

{a(n)=if(n==0,1,polcoeff(1-sum(m=0, n-1, a(m)*x^m*(1-(m+1)*x+x*O(x^n))^2), n))} /* Paul D. Hanna, Nov 27 2012 */

my(x='x+O('x^22)); Vec(serlaplace((1/(1-x))*exp(x/(1-x)))) \\ Joerg Arndt, Aug 11 2022

from math import factorial, comb
def A002720(n): return sum(factorial(k)*comb(n,k)**2 for k in range(n+1)) # Chai Wah Wu, Aug 31 2023

[factorial(n)*laguerre(n, -1) for n in (0..25)] # G. C. Greubel, Aug 11 2022

a(n) = Sum_{k=0..n} k!*C(n, k)^2.
E.g.f.: (1/(1-x))*exp(x/(1-x)). - Don Knuth, Jul 1995
D-finite with recurrence: a(n) = 2*n*a(n-1) - (n-1)^2*a(n-2).
a(n) = Sum_{k>=0} (k+n)! / ((k!)^2*exp(1)). - Robert G. Wilson v, May 02 2002 [corrected by Vaclav Kotesovec, Aug 28 2012]
a(n) = Sum_{m>=0} (-1)^m*A021009(n, m). - Philippe Deléham, Mar 10 2004
a(n) = Sum_{k=0..n} C(n, k)n!/k!. - Paul Barry, May 07 2004
a(n) = Sum_{k=0..n} P(n, k)*C(n, k); a(n) = Sum_{k=0..n} n!^2/(k!*(n-k)!^2). - Ross La Haye, Sep 20 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*Bell(k+1). - Vladeta Jovovic, Mar 18 2005
Define b(n) by b(0) = 1, b(n) = b(n-1) + (1/n) * Sum_{k=0..n-1} b(k). Then b(n) = a(n)/n!. - Franklin T. Adams-Watters, Sep 05 2005
Asymptotically, a(n)/n! ~ (1/2)*Pi^(-1/2)*exp(-1/2 + 2*n^(1/2))/n^(1/4) and so a(n) ~ C*BesselI(0, 2*sqrt(n))*n! with C = exp(-1/2) = 0.6065306597126334236... - Alec Mihailovs, Sep 06 2005, establishing a conjecture of Franklin T. Adams-Watters
a(n) = (n!/e) * Sum_{k>=0} binomial(n+k,n)/k!. - Gottfried Helms, Nov 25 2006
Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem): a(n) = Integral_{x=0..oo} x^n*BesselI(0,2*sqrt(x))*exp(-x)/exp(1) dx, n >= 0. - Karol A. Penson and G. H. E. Duchamp (gduchamp2(AT)free.fr), Jan 09 2007
a(n) = n! * LaguerreL[n, -1].
E.g.f.: exp(x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. - Paul D. Hanna, Nov 18 2011
From Peter Bala, Oct 11 2012: (Start)
Denominators in the sequence of convergents coming from Stieltjes's continued fraction for A073003, the Euler-Gompertz constant G := Integral_{x = 0..oo} 1/(1+x)*exp(-x) dx:
G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The numerators are in A002793. (End)
G.f.: 1 = Sum_{n>=0} a(n) * x^n * (1 - (n+1)*x)^2. - Paul D. Hanna, Nov 27 2012
E.g.f.: exp(x/(1-x))/(1-x) = G(0)/(1-x) where G(k) = 1 + x/((2*k+1)*(1-x) - x*(1-x)*(2*k+1)/(x + (1-x)*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 28 2012
a(n) = Sum_{k=0..n} L(n,k)*(k+1); L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014
a(n) = n! * A160617(n)/A160618(n). - Alois P. Heinz, Jun 28 2017
0 = a(n)*(-24*a(n+2) +99*a(n+3) -78*a(n+4) +17*a(n+5) -a(n+6)) +a(n+1)*(-15*a(n+2) +84*a(n+3) -51*a(n+4) +6*a(n+5)) +a(n+2)*(-6*a(n+2) +34*a(n+3) -15*a(n+4)) +a(n+3)*(+10*a(n+3)) for all n>=0. - Michael Somos, Jul 31 2018
a(n) = Sum_{k=0..n} C(n,k)*k!*A000262(n-k). - Geoffrey Critzer, Jan 07 2023
a(n) = A000262(n+1) - n * A000262(n). - Werner Schulte, Mar 29 2024
a(n) = denominator of (1 + n/(1 + n/(1 + (n-1)/(1 + (n-1)/(1 + ... + 1/(1 + 1/(1))))))). See A000262 for the numerators. - Peter Bala, Feb 11 2025

2nd description from R. H. Hardin, Nov 1997

3rd description from Wouter Meeussen, Jun 01 1998

A007489 a(n) = Sum_{k=1..n} k!.

Original entry on oeis.org

0, 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913, 43954713, 522956313, 6749977113, 93928268313, 1401602636313, 22324392524313, 378011820620313, 6780385526348313, 128425485935180313, 2561327494111820313, 53652269665821260313, 1177652997443428940313

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Equals row sums of triangle A143122 starting (1, 3, 9, 33, ...). - Gary W. Adamson, Jul 26 2008
a(n) for n>=4 is never a perfect square. - Alexander R. Povolotsky, Oct 16 2008
Number of cycles that can be written in the form (j,j+1,j+2,...), in all permutations of {1,2,...,n}. Example: a(3)=9 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), (132) we have 3+2+2+1+1+0=9 such cycles. - Emeric Deutsch, Jul 14 2009
Conjectured to be the length of the shortest word over {1,...,n} that contains each of the n! permutations as a factor (cf. A180632) [see Johnston]. - N. J. A. Sloane, May 25 2013
The above conjecture has been disproven for n>=6. See A180632 and the Houston 2014 reference. - Dmitry Kamenetsky, Mar 07 2016
a(n) is also the number of compositions of n if cardinal values do not matter but ordinal rankings do. Since cardinal values do not matter, a sequence of k summands summing to n can be represented as (s(1),...,s(k)), where the s's are positive integers and the numbers in parentheses are the initial ordinal rankings. The number of compositions of these summands are equal to k!, with k ranging from 1 to n. - Gregory L. Simay, Jul 31 2016
When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the left. Compare array A211370 for circular shifts to the left in a broader sense. Compare sequence A001563 for circular shifts to the right. - Tilman Piesk, Apr 29 2017
Since a(n) = (1!+2!+3!+...+n!) = 3(1+3!/3+4!/3+...+n!/3) is a multiple of 3 for n>2, the only prime in this sequence is a(2) = 3. - Eric W. Weisstein, Jul 15 2017
Generalization of 2nd comment: a(n) for n>=4 is never a perfect power (A007916) (Chentzov link). - Bernard Schott, Jan 26 2023

			a(4) = 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33. - _Michael B. Porter_, Aug 03 2016

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Section B44, Springer 2010.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Carauleanu Marc, Table of n, a(n) for n = 0..212 (first 100 terms from T. D. Noe)
N. N. Chentzov, D. O. Shklarsky, and I. M. Yaglom, The USSR Olympiad Problem Book, Selected Problems and Theorems of Elementary Mathematics, problem 115, pp. 28 and 201-202, Dover publications, Inc., New York, 1993.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Robin Houston, Tackling the Minimal Superpermutation Problem, arXiv:1408.5108 [math.CO], 2014.
Nathaniel Johnston, The minimal superpermutation problem (2013)
Nathaniel Johnston, Non-uniqueness of minimal superpermutations, Discrete Math. 313 (2013), no. 14, 1553--1557. MR3047396
S. Legendre and P. Paclet, On the Permutations Generated by Cyclic Shift , J. Int. Seq. 14 (2011) # 11.3.2.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
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.
Eric Weisstein's World of Mathematics, Factorial
Eric Weisstein's World of Mathematics, Left Factorial
G. Xiao, Sigma Server, Operate on "n!"
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 5.
Index entries for sequences related to factorial numbers

Equals A003422(n+1) - 1.
Cf. A000142, A000670, A001597, A007916, A137593, A143122, A161128, A180632.
Column k=0 of A120695.

List([1..20],n->Sum([1..n],Factorial)); # Muniru A Asiru, Jan 31 2018

a007489 n = a007489_list !! n
a007489_list = scanl (+) 0 $ tail a000142_list
-- Reinhard Zumkeller, Aug 29 2014

[0] cat [&+[Factorial(i): i in [1..n]]: n in [1..25]]; // Vincenzo Librandi, Sep 02 2016

A007489 := proc(n) local i; add(i!,i=1..n); end proc;

FoldList[Plus, 0, (Range@ 21)! ] (* Robert G. Wilson v, Sep 21 2007 *)
Table[Sum[i!, {i, 1, n}], {n, 0, 21}] (* Zerinvary Lajos, Jul 12 2009 *)
Accumulate[Range[50]!]  (* Harvey P. Dale, Apr 30 2011 *)
Table[Plus@@(Range[n]!), {n, 20}] (* Alonso del Arte, Jul 18 2011 *)

a(n)=sum(k=1,n,k!) \\ Charles R Greathouse IV, Jul 25 2011

a(n) = Sum_{k=1..n} P(n, k)/C(n, k). - Ross La Haye, Sep 21 2004
a(n) = 3*A056199(n) for n>=2. - Philippe Deléham, Feb 10 2007
a(n) = !(n+1)-1=A003422(n+1)-1. - Artur Jasinski, Nov 08 2007 [corrected by Werner Schulte, Oct 20 2021]
Starting (1, 3, 9, 33, 153, ...), = row sums of triangle A137593 - Gary W. Adamson, Jan 28 2008
a(n) = a(n-1) + n! for n >= 1. - Jaroslav Krizek, Jun 16 2009
E.g.f. A(x) satisfies to the differential equation A'(x)=A(x)+x/(1-x)^2+1. - Vladimir Kruchinin, Jan 22 2011
a(0)=0, a(1)=1, a(n) = (n+1)*a(n-1)-n*a(n-2). - Sergei N. Gladkovskii, Jul 05 2012
G.f.: W(0)*x/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
G.f.: x /(1-x)/Q(0),m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
E.g.f.: exp(x-1)*(Ei(1) - Ei(1-x)) - exp(x) + 1/(1 - x), where Ei(x) is the exponential integral. - Ilya Gutkovskiy, Nov 27 2016
a(n) = sqrt(a(n-1)*a(n+1)-a(n-2)*n*n!), n >= 2. - Gary Detlefs, Oct 26 2020
a(n) ~ n!. - Ridouane Oudra, Jun 11 2025

A033312 a(n) = n! - 1.

Original entry on oeis.org

0, 0, 1, 5, 23, 119, 719, 5039, 40319, 362879, 3628799, 39916799, 479001599, 6227020799, 87178291199, 1307674367999, 20922789887999, 355687428095999, 6402373705727999, 121645100408831999, 2432902008176639999, 51090942171709439999, 1124000727777607679999

Offset: 0

N. J. A. Sloane. This sequence appeared in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Eric W. Weisstein. Entry revised by N. J. A. Sloane, Jun 12 2012

a(n) gives the index number in any table of permutations of the entry in which the last n + 1 items are reversed. - Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004
a(n), n >= 1, has the factorial representation [n - 1, n - 2, ..., 1, 0]. The (unique) factorial representation of a number m from {0, 1, ... n! - 1} is m = sum(m_j(n)*j!, j = 0 .. n - 1) with m_j(n) from {0, 1, .., j}, n>=1. This is encoded as [m_{n-1},m_{n-2},...,m+1,m_0] with m_0=0. This can be interpreted as (D. N.) Lehmer code for the lexicographic rank of permutations of the symmetric group S_n (see the W. Lang link under A136663). The Lehmer code [n - 1, n - 2, ..., 1, 0] stands for the permutation [n, n - 1, ..., 1] (the last in lexicographic order). - Wolfdieter Lang, May 21 2008
For n >= 3: a(n) = numbers m for which there is one iteration {floor (r / k)} for k = n, n - 1, n - 2, ... 2 with property r mod k = k - 1 starting at r = m. For n = 5: a(5) = 119; floor (119 / 5) = 23, 119 mod 5 = 4; floor (23 / 4) = 5, 23 mod 4 = 3; floor (5 / 3) = 1, 5 mod 3 = 2; floor (1 / 2) = 0; 1 mod 2 = 1. - Jaroslav Krizek, Jan 23 2010
For n = 4, define the sum of all possible products of 1, 2, 3, 4 to be 1 + 2 + 3 + 4 add 1*2 + 1*3 + 1*4 add 2*3 + 2*4 + 3*4 add 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4 add 1*2*3*4. The sum of this is 119 = (4 + 1)! - 1. For n = 5 I get the sum 719 = (5 + 1)! - 1. The proof for the general case seems to follow by induction. - J. M. Bergot, Jan 10 2011

			G.f. = x^2 + 5*x^3 + 23*x^4 + 119*x^5 + 719*x^6 + 5039*x^7 + 40319*x^8 + ...

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 181, p. 92.
Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 6, 1969, p. 3, 1993.
Problem 598, J. Rec. Math., 11 (1978), 68-69.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
The IMO Compendium, Problem 6, 1st Canadian Mathematical Olympiad 1969.
Stéphane Legendre and Philippe Paclet, On the Permutations Generated by Cyclic Shift , J. Int. Seq. 14 (2011) # 11.3.2.
Gerard P. Michon, Wilson's Theorem.
Hisanori Mishima, Factorizations of many number sequences.
Hisanori Mishima, Factorizations of many number sequences.
Michael Penn, Make it look like a simple calculus problem., YouTube video, 2021.
Andrew Walker, Factors of n! +- 1.
Eric Weisstein's World of Mathematics, Factorial.
Eric Weisstein's World of Mathematics, Permutation Pattern.
Index entries for sequences related to factorial base representation.
Index entries for sequences related to factorial numbers.
Index to sequences related to Olympiads.

Cf. A000142, A001563 (first differences), A002582, A002982, A038507 (factorizations), A054415, A056110, A331373.

Row sums of A008291.

[Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

FoldList[#1*#2 + #2 - 1 &, 0, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
Range[0, 19]! - 1 (* Alonso del Arte, Jan 24 2013 *)

A033312(n):= n!-1$
makelist(A033312(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=n!-1 \\ Charles R Greathouse IV, Jul 19 2011

a(n) = Sum_{k = 1 .. n} (k-1)*(k-1)!.
a(n) = a(n - 1)*(n - 1) + a(n - 1) + n - 1, a(0) = 0. - Reinhard Zumkeller, Feb 03 2003
a(0) = a(1) = 0, a(n) = a(n - 1) * n + (n - 1) for n >= 2. - Jaroslav Krizek, Jan 23 2010
E.g.f.: 1/(1 - x) - exp(x). - Sergei N. Gladkovskii, Jun 29 2012
0 = 1 + a(n)*(+a(n+1) - a(n+2)) + a(n+1)*(+3 + a(n+1)) + a(n+2)*(-1) for n>=0. - Michael Somos, Feb 24 2017
Sum_{n>=2} 1/a(n) = A331373. - Amiram Eldar, Nov 11 2020

A163931 Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1.

Original entry on oeis.org

0, 9, 7, 8, 4, 3, 1, 9, 7, 2, 1, 6, 6, 7, 0, 1, 7, 9, 3, 2, 5, 5, 3, 7, 7, 8, 9, 0, 4, 5, 2, 8, 0, 0, 8, 2, 7, 6, 9, 5, 8, 2, 2, 6, 9, 5, 3, 0, 2, 6, 5, 7, 6, 5, 5, 7, 4, 4, 2, 1, 2, 4, 2, 4, 5, 4, 4, 7, 1, 3, 7, 6, 2, 6, 1, 4, 0, 9, 0, 4, 8, 8, 7, 3, 6, 9, 6, 0, 4, 8, 9, 1, 8, 5, 5, 5, 0, 8, 9, 4, 5, 4, 6, 7, 0

Offset: 0

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

We define the higher-order exponential integrals by E(x,m,n) = x^(n-1)*Integral_{t=x..infinity} E(t,m-1,n)/t^n for m >= 1 and n >= 1 with E(x,m=0,n) = exp(-x), see Meijer and Baken.
The properties of the E(x,m,n) are analogous to those of the well-known exponential integrals E(x,m=1,n), see Abramowitz and Stegun and the formulas.
The series expansions of the higher-order exponential integrals are dominated by the constants alpha(k,n), see A163927, and gamma(k,n) = G(k,n), see A090998.
For information about the asymptotic expansion of the E(x,m,n) see A163932.
Values of E(x,m,n) can be evaluated with the Maple program.

			E(1,2,1) = 0.09784319721667017932553778904528008276958226953026576557442124245....

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251.
J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.
M. S. Milgram, The generalized integro-exponential function, Math. of Computation, Vol. 44, pp. 443-458, 1985.
Eric Weisstein's World of Mathematics, The Exponential Integral.

Cf. A163927 (alpha(k,n)), A090998 (gamma(k,n) = G(k,n)), A163932.
Cf. A068985 (E(x=1,m=0,n) = exp(-1)) and A099285 (E(x=1,m=1,n=1)).
Cf. A001563 (n*n!), A002775 (n^2*n!), A091363 (n^3*n!) and A091364 (n^4*n!).

E:= proc(x,m,n) local nmax, kmax, EI, k1, k2, n1, n2; option remember: nmax:=20; kmax:=20; k1:=0: for n1 from 0 to nmax do alpha(k1,n1):=1 od: for k1 from 1 to kmax do for n1 from 1 to nmax do alpha(k1,n1) := (1/k1)*sum(sum(p^(-2*(k1-i1)),p=0..n1-1)*alpha(i1, n1),i1=0..k1-1) od; od: for n2 from 0 to kmax do G(0,n2):=1 od: for n2 from 1 to nmax do for k2 from 1 to kmax do G(k2,n2):=(1/k2)*(((gamma-sum(p^(-1),p=1..n2-1))*G(k2-1,n2)+ sum((Zeta(k2-i2)-sum(p^(-(k2-i2)), p=1..n2-1))*G(i2,n2),i2=0..k2-2))) od; od: EI:= evalf((-1)^m*((-x)^(n-1)/(n-1)!*sum(alpha(kz,n)*(G(m-2*kz,n)+sum(G(m-2*kz-i,n)*ln(x)^i/i!,i=1..m-2*kz)), kz=0..floor(m/2)) + sum((-x)^kx/((kx-n+1)^m*kx!),kx=0..n-2) + sum((-x)^ky/((ky-n+1)^m*ky!),ky=n..infinity))); return(EI): end:

Join[{0}, RealDigits[ N[ EulerGamma^2/2 + Pi^2/12 - HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -1], 104]][[1]]] (* Jean-François Alcover, Nov 07 2012, from 1st formula *)

t=1; Euler^2/2 + Pi^2/12 + sumalt(k=1, t*=k; (-1)^k/(k^2*t)) \\ Charles R Greathouse IV, Nov 07 2016

E(x=1,m=2,n=1) = gamma^2/2 + Pi^2/12 + Sum_{k>=1} ((-1)^k/(k^2*k!)).
E(x=0,n,m) = (1/(n-1))^m for n >= 2.
Integral_{t=0..x} E(t,m,n) = 1/n^m - E(x,n,n+1).
dE(x,m,n+1)/dx = - E(x,m,n).
E(x,m,n+1) = (1/n)*(E(x,m-1,n+1) - x*E(x,m,n)).
E(x,m,n) = (-1)^m * ((-x)^(n-1)/(n-1)!) * Sum_{kz=0..floor(m/2)}(alpha (kz, n)*G(m-2*kz, n)) + (-1) ^m * ((-x)^(n-1)/(n-1)!) * Sum_{kz=0..floor(m/2)}(Sum_{i=1..m-2*kz}(alpha (kz, n) *G(m-2*kz-i, n)*log(x)^i/i!)) + (-1)^m * Sum_{ kx=0..n-2}((-x)^kx/((kx-n+1)^m*kx!) + (-1)^m * Sum_{ky>=n}((-x)^ky /(( ky-n+1)^m*ky!)).

A001339 a(n) = Sum_{k=0..n} (k+1)! binomial(n,k).

Original entry on oeis.org

1, 3, 11, 49, 261, 1631, 11743, 95901, 876809, 8877691, 98641011, 1193556233, 15624736141, 220048367319, 3317652307271, 53319412081141, 909984632851473, 16436597430879731, 313262209859119579, 6282647653285676001, 132266266384961600021, 2916471173788403280463

Offset: 0

N. J. A. Sloane

nonn

Number of arrangements of {1, 2, ..., n, n + 1} containing the element 1. - Emeric Deutsch, Oct 11 2001
From Thomas Wieder, Oct 21 2004: (Start)
"Also the number of hierarchies with unlabeled elements and labeled levels where the levels are permuted.
"Let l_x denote level x, e.g. l_2 is level 2. Let * denote an element. Then l_1*l_2***l_3** denotes a hierarchy of n = 6 unlabeled elements with one element on level 1, three elements on level 2 and 2 elements on level 3.
"E.g. for n=3 one has a(3) = 11 possible hierarchies: l_1***, l_1**l_2*, l_1*l_2**, l_2**l_1*, l_2*l_1**, l_1*l_2*l_3*, l_3*l_1*l_2*, l_2*l_3*l_1*, l_1*l_3*l_2*, l_2*l_1*l_3*, l_3*l_2*l_1*. See A064618 for the number of hierarchies with labeled elements and labeled levels." (End)
Polynomials in A010027 evaluated at 2.
Also the permanent of any n X n cofactor of an n+1 X n+1 version of J+I other than an n X n version of J + I (that is, a (1, 2) matrix with n - 1 2s, at most one per row and column). - D. G. Rogers, Aug 27 2006
a(n) = number of partitions of [n+1] into lists of sets that are both non-nesting and non-crossing. Non-nesting means that no set is contained in the span (interval from min to max) of another. For example, a(1) counts 12, 1-2, 2-1 and a(2) counts 123, 1-23, 23-1, 3-12, 12-3, 1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2, 3-2-1. - David Callan, Sep 20 2007
Row sums of triangle A137594. - Gary W. Adamson, Jan 28 2008
From Peter Bala, Jul 10 2008: (Start)
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). See A000522 for further properties of difference divisibility sequences.
a(n) equals the sum of the lengths of the paths between a pair of distinct vertices of the complete graph K_(n + 2) on n + 2 vertices [Hassani]. For example, for the complete graph K_4 with vertex set {A,B,C,D} the 5 paths between A and B are AB of length 1, ACB and ADB, both of length 2 and ACDB and ADCB, both of length 3. The sum of the lengths is 1 + 2 + 2 + 3 + 3 = 11 = a(2).
The number of paths between 2 distinct vertices of K_n is equal to A000522(n - 2); the number of simple cycles through a vertex of K_n equals A038154(n - 1).
Recurrence relation: a(0) = 1, a(1) = 3, a(n) = (n+2)*a(n - 1) - (n - 1)*a(n - 2) for n >= 2. The sequence b(n) := n*n! = A001563(n) satisfies the same recurrence with the initial conditions b(0) = 0, b(1) = 1. This leads to the finite continued fraction expansion a(n)/b(n) = 3 - 1/(4 - 2/(5 - 3/(6 - ... - (n - 1)/(n + 2)))), n >= 1.
Limit_{n->oo} a(n)/b(n) = e = 3 - 1/(4 - 2/(5 - 3/(6 - ... - n/((n + 3) - ...)))).
For n >= 1, a(n) = b(n)*(3 - Sum_{k=2..n} 1/(k!*(k - 1)*k)) (see the formula by Deutsch) since the rhs satisfies the above recurrence with the same initial conditions. Hence e = 3 - Sum_{k>=2} 1/(k!*(k - 1)*k).
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 A000522 (r=0), A082030 (r=2), A095000 (r=3) and A095177 (r=4). (End)
Binomial transform of n! Offset 1. a(3) = 11. - Al Hakanson (hawkuu(AT)gmail.com), May 18 2009
Equals eigensequence of a triangle with (1, 2, 3, ...) as the right border and the rest 1's; equivalent to a(n) = [n terms of the sequence (1, 1, 1, ...) followed by (n + 1)] dot [(n + 1) terms of the sequence (1, 1, 3, 11, 245, ...)]. Example: 261 = a(4) = (1, 1, 1, 1, 5) dot (1, 1, 3, 11, 49) = 1 + 1 + 3 + 11 + 245 = 261. - Gary W. Adamson, Jul 24 2010
Number of nonnegative integers which use each digit at most once in base n+1. - Franklin T. Adams-Watters, Oct 02 2011
a(n) is the number of permutations of {1,2,...,n+2} in which there is an increasing contiguous subsequence (ascending run) beginning with 1 and ending with n+2. The number of such permutations with exactly k, 0<=k<=n, elements between the 1 and (n+2) is given by A132159(n,k) whose row sums equal this sequence. See example. - Geoffrey Critzer, Feb 15 2013

			G.f. = 1 + 3*x + 11*x^2 + 49*x^3 + 261*x^4 + 1631*x^5 + 11743*x^6 + 95901*x^7 + ...
a(2) = 11: {1, 12, 21, 13, 31, 123, 132, 213, 231, 312, 321}.
a(2) = 11 because we have 11 permutations of {1,2,3,4} (written in one line notation) that have an increasing subsequence beginning with 1 and ending with 4: 1,2,3,4; 1,2,4,3; 1,3,4,2; 1,4,2,3; 1,4,3,2; 2,1,3,4; 2,1,4,3; 2,3,1,4; 3,1,2,4; 3,1,4,2; 3,2,1,4. - _Geoffrey Critzer_, Feb 15 2013

A. Hordijk, Markov Decision Chains, pp. 97-103 in Images of SMC Research, 1996, Stichting Mathematisch Centrum, Amsterdam, Netherlands, 1996. See p. 103.
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).
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 56, ex. 232.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas, Freeman, San Francisco, 1974. [Annotated scans of pages 69 and 70 only]
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7.
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
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.
Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson Scheme, and Generalized Derangements, International Journal of Combinatorics, Volume 2011, Article ID 539030, 29 pages.
Loïc Foissy and Frédéric Patras, Natural endomorphisms of shuffle algebras, arXiv preprint arXiv:1205.2986 [math.RA], 2012.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
M. Hassani, Counting and computing by e, arXiv:math/0606613 [math.CO], 2006.
F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 400
M. J. Knight and W. O. Egerland, Solution to Problem 5911, Amer. Math. Monthly 81 (1974) 675-676.
Zhentao Lu, Elementary proofs of generalized continued fraction formulae for e, arXiv:1907.05563 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.

Cf. A001563, A064618, A089900, A137594.
a(n) = A000522(n+1) - A000522(n).
First differences of A000522, A007526, A026243, A073591.
Equals (1/2)*A006183(n-2).
Equals A036918(n+1) + 1.
Leftmost column of A276588.
Cf. also A136104.

A001339:=List([0..20],n-> Sum([0..n], k-> Factorial(k+1)*Binomial(n,k))); # Muniru A Asiru, Feb 17 2018

[Factorial(n)*(&+[(n-k+1)/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 15 2019

a:=proc(n) options operator, arrow: factorial(n)*n*(3-(sum(1/(j*(j-1)*factorial(j)), j=2..n))) end proc: 1, seq(a(n),n=1..20); # Emeric Deutsch, Apr 12 2008
a := n -> hypergeom([2, -n], [], -1); seq(simplify(a(n)), n=0..18); # Peter Luschny, Sep 20 2014

a[n_] := n!*Sum[(k+1)/(n-k)!, {k, 0, n}]; a /@ Range[0, 20] (* Jean-François Alcover, Jul 13 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[x] / (1 - x)^2, {x, 0, n}]] (* Michael Somos, Oct 20 2011 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp(x + x * O(x^n)) / (1 - x)^2, n))} /* Michael Somos, Mar 04 2004 */

vector(20, n, n--; n!*sum(k=0,n,(n-k+1)/k!)) \\ G. C. Greubel, Jul 15 2019

[factorial(n)*sum((n-k+1)/factorial(k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 15 2019

E.g.f.: exp(x)/(1-x)^2.
a(n) = round(evalf(exp(1)*(n-1)*(n-1)!)) (n>1).
a(n) = floor(n*n!*e) + 1. - Melvin J. Knight (knightmj(AT)juno.com), May 30 2001
a(n) = {e*n*n!} for n > 0, where {x} denotes the nearest integer part. Proposed by Simon Plouffe, March 1993.
The n-th row of array A089900 is the n-th binomial transform of this sequence. The (n+1)-th term of the n-th binomial transform is (n+1)^(n+1), for n >= 0. E.g., the 5th term of the 4th binomial transform is 5^5: [1, 7, 51, 389, 3125, ...]. - Paul D. Hanna, Nov 14 2003
G.f.: Sum_{k>=0} k! * (x / (1 - x))^k. - Michael Somos, Mar 04 2004
a(n) = Sum_{k = 0..n} A046716(n, k)*2^(n-k). - Philippe Deléham, Jun 12 2004
(n-1)*a(n) = n^2*a(n-1)-1. - Vladeta Jovovic, Sep 04 2004
a(n) = Sum_{k=0..n} P(n, k)*(k+1). - Ross La Haye, Aug 28 2005
a(n) = n!*n*(3 - Sum_{j=2..n} 1/(j*(j-1)*j!)) for n>=1. - Emeric Deutsch, Apr 12 2008
a(n) = (a(n-1)^2 + 2 * a(n-2)^2 + a(n-2) * a(n-3) - 4 * a(n-1) * a(n-3)) / (a(n-2) - a(n-3)) if n>1. - Michael Somos, Oct 20 2011
E.g.f.:1/Q(0); Q(k) = 1 - 2*x/(1+x/(2-x-2/(1-x*(k+1)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011
G.f.: 1/Q(0), where Q(k) = 1 - x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: Q(0)/x - 1/x, where Q(k) = 1 + (2*k + 1)*x/( 1 - x - 2*x*(1-x)*(k+1)/(2*x*(k+1) + (1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 09 2013
G.f.: (2/x)/G(0) - 1/x, 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.: Q(0)/(2*x) - 1/x, where 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.: W(0)/x - 1/x, where W(k) = 1 - x*(k+1)/( x*(k+2) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013
a(n) = hypergeometric([2, -n], [], -1). - Peter Luschny, Sep 20 2014
Upper and bottom right terms of the infinite 2 X 2 matrix product_{N=1,2,3,...} [(1,1); (1,N)]. - Gary W. Adamson, Jul 28 2016
a(n) = R(n,n+1,n) where R(x,y,z) is defined to be R(x+1,y,z+1) = R(y,x,x) + R(z,y,z), R(0,y,z+1) = R(z,y,z), R(x+1,y,0) = R(y,x,x), and R(0,y,0) = y. - David M. Cerna, Feb 16 2018
a(n) = (n + 1)!*hypergeom([-n], [-n-1], 1). - Peter Luschny, Nov 02 2018
a(n) = Integral_{x=0..1} (-LambertW(-1,-x/e))^n dx. - Gleb Koloskov, Jul 25 2021
a(n) = KummerU(-n, -n-1, 1). - Peter Luschny, May 10 2022

Typo in description in 1995 Encyclopedia of Integer Sequences corrected Mar 15 1997

Link updated by Susanne Wienand, Nov 19 2011

A055089 List of all finite permutations in reversed colexicographic ordering.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 3, 1, 2, 2, 3, 1, 3, 2, 1, 1, 2, 4, 3, 2, 1, 4, 3, 1, 4, 2, 3, 4, 1, 2, 3, 2, 4, 1, 3, 4, 2, 1, 3, 1, 3, 4, 2, 3, 1, 4, 2, 1, 4, 3, 2, 4, 1, 3, 2, 3, 4, 1, 2, 4, 3, 1, 2, 2, 3, 4, 1, 3, 2, 4, 1, 2, 4, 3, 1, 4, 2, 3, 1, 3, 4, 2, 1, 4, 3, 2, 1, 1, 2, 3, 5, 4, 2, 1, 3, 5, 4, 1, 3, 2, 5, 4, 3, 1, 2

Offset: 0

Antti Karttunen, Apr 18 2000

			In this table, each row consists of A001563(n) permutations of n+1 terms; i.e., we have (1/) 2,1/ 1,3,2; 3,1,2; 2,3,1; 3,2,1/ 1,2,4,3; 2,1,4,3; ... .
Append to each an infinite number of fixed terms and we get a list of rearrangements of the natural numbers, but with only a finite number of terms permuted:
1/2,3,4,5,6,7,8,9,...
2,1/3,4,5,6,7,8,9,...
1,3,2/4,5,6,7,8,9,...
3,1,2/4,5,6,7,8,9,...
2,3,1/4,5,6,7,8,9,...
3,2,1/4,5,6,7,8,9,...
1,2,4,3/5,6,7,8,9,...
2,1,4,3/5,6,7,8,9,...
Alternatively, if we take only the first n terms of each such infinite row, then the first n! rows give all permutations of the elements 1,2,...,n.

Antti Karttunen, Rows 0..719 of the irregular table, flattened.
Daniel Forgues, Tilman Piesk, et al., Orderings, OEIS Wiki.
Antti Karttunen, Ranking and unranking functions, OEIS Wiki.
Tilman Piesk, Permutations and partitions in the OEIS, Wikiversity.
Lorenzo Sauras-Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.
Index entries for sequences related to factorial base representation
Index entries for sequences related to permutations

Inversion vectors: A007623, cycle counts: A055090, minimum number of transpositions: A055091, minimum number of adjacent transpositions: A034968, order of each permutation: A055092, number of non-fixed elements: A055093, positions of inverses: A056019, positions after Foata transform: A065181; positions of fixed-point-free involutions: A064640.
Cf. A057112, A060112, A060117, A060118, A060132, A060133.
Cf. A195663, array of the infinite rows.
This permutation list gives essentially the same information as A030298/A030299, but in a more compact way, by skipping those permutations of A030298 that start with a fixed element.
A220658(n) gives the rank r of the permutation of which the term at a(n) is an element.
A220659(n) gives the zero-based position (from the left) of that a(n) in that permutation of rank r.
A084558(r)+1 gives the size of the finite subsequence (of the r-th infinite, but finitary permutation) which has been included in this list.

factorial_base := proc(nn) local n,a,d,j,f; n := nn; if(0 = n) then RETURN([0]); fi; a := []; f := 1; j := 2; while(n > 0) do d := floor(`mod`(n,(j*f))/f); a := [d,op(a)]; n := n - (d*f); f := j*f; j := j+1; od; RETURN(a); end;
fexlist2permlist := proc(a) local n,b,j; n := nops(a); if(0 = n) then RETURN([1]); fi; b := fexlist2permlist(cdr(a)); for j from 1 to n do if(b[j] >= ((n+1)-a[1])) then b[j] := b[j]+1; fi; od; RETURN([op(b),(n+1)-a[1]]); end;
fac_base := n -> fac_base_aux(n,2); fac_base_aux := proc(n,i) if(0 = n) then RETURN([]); else RETURN([op(fac_base_aux(floor(n/i),i+1)), (n mod i)]); fi; end;
PermRevLexUnrank := n -> `if`((0 = n),[1],fexlist2permlist(fac_base(n)));
cdr := proc(l) if 0 = nops(l) then ([]) else (l[2..nops(l)]); fi; end; # "the tail of the list"
# Same algorithm in different guise, showing how permutations are composed of adjacent transpositions (compare to algorithm PermUnrank3R at A060117):
PermRevLexUnrankAMSDaux := proc(n,r, pp) local s,p,k; p := pp; if(0 = r) then RETURN(p); else s := floor(r/((n-1)!)); for k from n-s to n-1 do p := permul(p,[[k,k+1]]); od; RETURN(PermRevLexUnrankAMSDaux(n-1, r-(s*((n-1)!)), p)); fi; end;
PermRevLexUnrankAMSD := proc(r) local n; n := nops(factorial_base(r)); convert(PermRevLexUnrankAMSDaux(n+1,r,[]),'permlist',1+(((r+2) mod (r+1))*n)); end;

A055089L[n_] := Reverse@SortBy[DeleteCases[Permutations@Range@n, {, n}], Reverse]; Flatten@Array[A055089L, 4] (* JungHwan Min, Aug 28 2016 *)

[seq(op(PermRevLexUnrank(j)), j=0..)]; (see Maple code given below).

Name changed by Tilman Piesk, Feb 01 2012

A084558 a(0) = 0; for n >= 1: a(n) = largest m such that n >= m!.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5

Offset: 0

Antti Karttunen, Jun 23 2003

For n >= 1, a(n) = the number of significant digits in n's factorial base representation (A007623).
After zero, which occurs once, each n occurs A001563(n) times.
Number of iterations (...(f_4(f_3(f_2(n))))...) such that the result is < 1, where f_j(x):=x/j. - Hieronymus Fischer, Apr 30 2012
For n > 0: a(n) = length of row n in table A108731. - Reinhard Zumkeller, Jan 05 2014

			a(4) = 2 because 2! <= 4 < 3!.

F. Smarandache, "f-Inferior and f-Superior Functions - Generalization of Floor Functions", Arizona State University, Special Collections.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Yi Yuan and Zhang Wenpeng, On the Mean Value of the Analogue of Smarandache Function, Smarandache Notions J., Vol. 15.

A dual to A090529.
Cf. A084555, A084556, A084557.
Cf. A001069, A010096.
Cf. A000142, A001563, A055089, A060130, A111095, A211664, A211670, A108731, A212598, A220656, A220657, A220658, A220659, A231716, A235224, A257510.

a084558 n = a090529 (n + 1) - 1  -- Reinhard Zumkeller, Jan 05 2014

0, seq(m$(m*m!),m=1..5); # Robert Israel, Apr 27 2015

Table[m = 1; While[m! <= n, m++]; m - 1, {n, 0, 104}] (* Jayanta Basu, May 24 2013 *)
Table[Floor[Last[Reduce[x! == n && x > 0, x]]], {n, 120}] (* Eric W. Weisstein, Sep 13 2024 *)

a(n)={my(m=0);while(n\=m++,);m-1} \\ R. J. Cano, Apr 09 2018

def A084558(n):
  i=1
  while n: i+=1; n//=i
  return(i-1)
print(list(map(A084558,range(101)))) # Natalia L. Skirrow, May 28 2023

From Hieronymus Fischer, Apr 30 2012: (Start)
a(n!) = a((n-1)!)+1, for n>1.
G.f.: 1/(1-x)*Sum_{k>=1} x^(k!).
The explicit first terms of the g.f. are: (x+x^2+x^6+x^24+x^120+x^720...)/(1-x).
(End)
Other identities:
For all n >= 0, a(n) = A090529(n+1) - 1. - Reinhard Zumkeller, Jan 05 2014
For all n >= 1, a(n) = A060130(n) + A257510(n). - Antti Karttunen, Apr 27 2015
a(n) ~ log(n^2/(2*Pi)) / (2*LambertW(log(n^2/(2*Pi))/(2*exp(1)))) - 1/2. - Vaclav Kotesovec, Aug 22 2025

Name clarified by Antti Karttunen, Apr 27 2015

A225901 Write n in factorial base, then replace each nonzero digit d of radix k with k-d.

Original entry on oeis.org

0, 1, 4, 5, 2, 3, 18, 19, 22, 23, 20, 21, 12, 13, 16, 17, 14, 15, 6, 7, 10, 11, 8, 9, 96, 97, 100, 101, 98, 99, 114, 115, 118, 119, 116, 117, 108, 109, 112, 113, 110, 111, 102, 103, 106, 107, 104, 105, 72, 73, 76, 77, 74, 75, 90, 91, 94, 95, 92, 93, 84, 85, 88, 89, 86, 87, 78, 79, 82, 83, 80, 81, 48, 49, 52, 53, 50, 51, 66, 67, 70, 71, 68

Offset: 0

Paul Tek, May 20 2013

Analogous to A004488 or A048647 for the factorial base.
A self-inverse permutation of the natural numbers.
From Antti Karttunen, Aug 16-29 2016: (Start)
Consider the following way to view a factorial base representation of nonnegative integer n. For each nonzero digit d_i present in the factorial base representation of n (where i is the radix = 2.. = one more than 1-based position from the right), we place a pebble to the level (height) d_i at the corresponding column i of the triangular diagram like below, while for any zeros the corresponding columns are left empty:
.
Level
  6        o
          ─ ─
  5        . .
          ─ ─ ─
  4        . . .
          ─ ─ ─ ─
  3        . . . .
          ─ ─ ─ ─ ─
  2        . . o . .
          ─ ─ ─ ─ ─ ─
  1        . o . . o o
          ─ ─ ─ ─ ─ ─ ─
  Radix:   7 6 5 4 3 2
  Digits:  6 1 2 0 1 1 = A007623(4491)
Instead of levels, we can observe on which "slope" each pebble (nonzero digit) is located at. Formally, the slope of nonzero digit d_i with radix i is (i - d_i). Thus in above example, both the most significant digit (6) and the least significant 1 are on slope 1 (called "maximal slope", because it contains digits that are maximal allowed in those positions), while the second 1 from the right is on slope 2 ("submaximal slope").
This involution (A225901) sends each nonzero digit at level k to the slope k (and vice versa) by flipping such a diagram by the shallow diagonal axis that originates from the bottom right corner. Thus, from above diagram we obtain:
Slope (= digit's radix - digit's value)
   1
   2 .
   3 .  .╲
   4 .  .╲o╲
   5 .  .╲.╲.╲
   6 .  .╲.╲o╲.╲
     .  .╲.╲.╲.╲o╲
        o╲.╲.╲.╲.╲o╲
        -----------------
        1  5  3 0  2  1  = A007623(1397)
and indeed, a(4491) = 1397 and a(1397) = 4491.
Thus this permutation maps between polynomial encodings A275734 & A275735 and all the respective sequences obtained from them, where the former set of sequences are concerned with the "slopes" and the latter set with the "levels" of the factorial base representation. See the Crossrefs section.
Sequences A231716 and A275956 are closed with respect to this sequence, in other words, for all n, a(A231716(n)) is a term of A231716 and a(A275956(n)) is a term of A275956.
(End)

			a(1000) = a(1*6! + 2*5! + 1*4! + 2*3! + 2*2!) = (7-1)*6! + (6-2)*5! + (5-1)*4! + (4-2)*3! + (3-2)*2! = 4910.
a(1397) = a(1*6! + 5*5! + 3*4! + 0*3! + 2*2! + 1*1!) = (7-1)*6! + (6-5)*5! + (5-3)*4! + (3-2)*2! + (2-1)*1! = 4491.

Cf. A000142, A007623, A004488, A048647, A001563, A007489, A257684, A257687, A276091, A275736, A276149.
Cf. A275959 (fixed points), A231716, A275956.
Cf. A153880 & A255411.
Cf. also A275734 & A275735, A275952 & A275954.
This involution maps between the following sequences related to "levels" and "slopes" (see comments): A275806 <--> A060502, A257511 <--> A260736, A264990 <--> A275811, A275729 <--> A275728, A275948 <--> A275946, A275949 <--> A275947, A275964 <--> A275962, A059590 <--> A276091.
Related permutations: A275957, A275958, A275835, A275836, A275837, A275838, A276011, A276012.

b = MixedRadix[Reverse@ Range[2, 12]]; Table[FromDigits[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #), b] &@ IntegerDigits[n, b], {n, 0, 82}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Range[# + 1] <= n &]; Most@ Rest[a][[All, 1]] /. {} -> {0}]; g[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Range@ Range[0, Length@ w]], Reverse@ Append[w, 0]}]; Table[g[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #)] &@ f@ n, {n, 0, 82}] (* Michael De Vlieger, Aug 29 2016 *)

a(n)=my(s=0,d,k=2);while(n,d=n%k;n=n\k;if(d,s=s+(k-d)*(k-1)!);k=k+1);return(s)

from sympy import factorial as f
def a(n):
    s=0
    k=2
    while(n):
        d=n%k
        n=(n//k)
        if d: s=s+(k - d)*f(k - 1)
        k+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

(define (A225901 n) (let loop ((n n) (z 0) (m 2) (f 1)) (cond ((zero? n) z) (else (loop (quotient n m) (if (zero? (modulo n m)) z (+ z (* f (- m (modulo n m))))) (+ 1 m) (* f m))))))
;; One implementing the first recurrence, with memoization-macro definec:
(definec (A225901 n) (if (zero? n) n (+ (A276091 (A275736 n)) (A153880 (A225901 (A257684 n))))))
;; Antti Karttunen, Aug 29 2016

From Antti Karttunen, Aug 29 2016: (Start)
a(0) = 0; for n >= 1, a(n) = A276091(A275736(n)) + A153880(a(A257684(n))).
or, for n >= 1, a(n) = A276149(n) + a(A257687(n)).
(End)
Other identities. For n >= 0:
a(n!) = A001563(n).
a(n!-1) = A007489(n-1).
From Antti Karttunen, Aug 16 2016: (Start)
A275734(a(n)) = A275735(n) and vice versa, A275735(a(n)) = A275734(n).
A060130(a(n)) = A060130(n). [The flip preserves the number of nonzero digits.]
A153880(n) = a(A255411(a(n))) and A255411(n) = a(A153880(a(n))). [This involution conjugates between the two fundamental factorial base shifts.]
a(n) = A257684(a(A153880(n))) = A266193(a(A255411(n))). [Follows from above.]
A276011(n) = A273662(a(A273670(n))).
A276012(n) = A273663(a(A256450(n))).
(End)

A021009 Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).

Original entry on oeis.org

1, 1, -1, 2, -4, 1, 6, -18, 9, -1, 24, -96, 72, -16, 1, 120, -600, 600, -200, 25, -1, 720, -4320, 5400, -2400, 450, -36, 1, 5040, -35280, 52920, -29400, 7350, -882, 49, -1, 40320, -322560, 564480, -376320, 117600, -18816, 1568, -64, 1, 362880, -3265920

Offset: 0

N. J. A. Sloane

In absolute values, this sequence also gives the lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals (j-1)^2 (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006
A partial permutation on a set X is a bijection between two subsets of X. |T(n,n-k)| equals the numbers of partial permutations of an n-set having domain cardinality equal to k. Let E denote the operator D*x*D, where D is the derivative operator d/dx. Then E^n = Sum_{k = 0..n} |T(n,k)|*x^k*D^(n+k). - Peter Bala, Oct 28 2008
The unsigned triangle is the generalized Riordan array (exp(x), x) with respect to the sequence n!^2 as defined by Wang and Wang (the generalized Riordan array (exp(x), x) with respect to the sequence n! is Pascal's triangle A007318, and with respect to the sequence n!*(n+1)! is A105278). - Peter Bala, Aug 15 2013
The unsigned triangle appears on page 83 of Ser (1933). - N. J. A. Sloane, Jan 16 2020

			The triangle a(n,m) starts:
n\m   0       1      2       3      4      5    6  7  8
0:    1
1:    1      -1
2:    2      -4      1
3:    6     -18      9      -1
4:   24     -96     72     -16      1
5:  120    -600    600    -200     25     -1
6:  720   -4320   5400   -2400    450    -36    1
7: 5040  -35280  52920  -29400   7350   -882   49  -1
8:40320 -322560 564480 -376320 117600 -18816 1568 -64 1
...
From _Wolfdieter Lang_, Jan 31 2013 (Start)
Recurrence (usual one): a(4,1) = 7*(-18) - 6 - 3^2*(-4) = -96.
Recurrence (simplified version): a(4,1) = 5*(-18) - 6 = -96.
Recurrence (Sage program): |a(4,1)| = 6 + 3*18 + 4*9 = 96. (End)
Embedded recurrence (Maple program): a(4,1) = -4!*(1 + 3) = -96.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 83.

T. D. Noe, Rows n = 0..50 of triangle, flattened
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].
W. A. Al-Salam, Operational representations for the Laguerre and other polynomials, Duke Math. Jour., vol 31 (1964), pp. 127-142.
Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 5.
Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 7.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 21.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011) # 11.6.7.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
A. Belov-Kanel and M. Kontsevich, Automorphisms of the Weyl algebra, arXiv preprint arXiv:0512169 [math.QA], 2005.
A. Belov-Kanel and M. Kontsevich, The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture, arXiv preprint arXiv:0512171 [math.RA], 2005.
I. Gessel, Applications of the classical umbral calculus
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See. p. 19.
Massimo Nocentini, An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation, PhD Thesis, University of Florence, 2019. See p. 31.
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)
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
W. Wang and T. Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Index entries for sequences related to Laguerre polynomials

Row sums give A009940, alternating row sums are A002720.
Column sequences (unsigned): A000142, A001563, A001809-A001812 for m=0..5.
Central terms: A295383.
For generators and generalizations see A132440.
Cf. A021010, A025166, A025167, A062137, A062138, A062139, A062140, A066667, A321620.

/* As triangle: */ [[((-1)^k)*Factorial(n)*Binomial(n, k)/Factorial(k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jan 18 2020

A021009 := proc(n,k) local S; S := proc(n,k) option remember; `if`(k = 0, 1, `if`( k > n, 0, S(n-1,k-1)/k + S(n-1,k))) end: (-1)^k*n!*S(n,k) end: seq(seq(A021009(n,k), k=0..n), n=0..8); # Peter Luschny, Jun 21 2017
# Alternative for the unsigned case (function RiordanSquare defined in A321620):
RiordanSquare(add(x^m, m=0..10), 10, true); # Peter Luschny, Dec 06 2018

Flatten[ Table[ CoefficientList[ n!*LaguerreL[n, x], x], {n, 0, 9}]] (* Jean-François Alcover, Dec 13 2011 *)

p(n) = denominator(bestapprPade(Ser(vector(2*n, k, (k-1)!))));
concat(1, concat(vector(9, n, Vec(-p(n)))))  \\ Gheorghe Coserea, Dec 01 2016

{T(n, k) = if( n<0, 0, n! * polcoeff( sum(i=0, n, binomial(n, n-i) * (-x)^i / i!), k))}; /* Michael Somos, Dec 01 2016 */

row(n) = Vecrev(n!*pollaguerre(n)); \\ Michel Marcus, Feb 06 2021

def A021009_triangle(dim): # computes unsigned T(n,k).
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(2*k+1)*M[n-1,k]+(k+1)^2*M[n-1,k+1]
    return M
A021009_triangle(9) # Peter Luschny, Sep 19 2012

a(n, m) = ((-1)^m)*n!*binomial(n, m)/m! = ((-1)^m)*((n!/m!)^2)/(n-m)! if n >= m, otherwise 0.
E.g.f. for m-th column: (-x/(1-x))^m /((1-x)*m!), m >= 0.
Representation (of unsigned a(n, m)) as special values of Gauss hypergeometric function 2F1, in Maple notation: n!*(-1)^m*hypergeom([ -m, n+1 ], [ 1 ], 1)/m!. - Karol A. Penson, Oct 02 2003
Sum_{m>=0} (-1)^m*a(n, m) = A002720(n). - Philippe Deléham, Mar 10 2004
E.g.f.: (1/(1-x))*exp(x*y/(x-1)). - Vladeta Jovovic, Apr 07 2005
Sum_{n>=0, m>=0} a(n, m)*(x^n/n!^2)*y^m = exp(x)*BesselJ(0, 2*sqrt(x*y)). - Vladeta Jovovic, Apr 07 2005
Matrix square yields the identity matrix: L^2 = I. - Paul D. Hanna, Nov 22 2008
From Tom Copeland, Oct 20 2012: (Start)
Symbolically, with D=d/dx and LN(n,x)=n!L_n(x), define :Dx:^j = D^j x^j, :xD:^j = x^j D^j, and LN(.,x)^j = LN(j,x) = row polynomials of A021009.
Then some useful relations are
1) (:Dx:)^n = LN(n,-:xD:)    [Rodriguez formula]
2) (xDx)^n = x^n D^n x^n = x^n LN(n,-:xD:)  [See Al-Salam ref./A132440]
3) (DxD)^n = D^n x^n D^n = LN(n,-:xD:) D^n  [See ref. in A132440]
4) umbral composition LN(n,LN(.,x))= x^n   [See Rota ref.]
5) umbral comp. LN(n,-:Dx:) = LN(n,-LN(.,-:xD:)) = 2^n LN(n,-:xD:/2)= n! * (n-th row e.g.f.(x) of A038207 with x replaced by :xD:).
An example for 2) is the operator (xDx)^2 = (xDx)(xDx) = xD(x^2 + x^3D)= 2x^2 + 4x^3 D + x^4 D^2 = x^2 (2 + 4x D + x^2 D^2) = x^2 (2 + 4 :xD: + :xD:^2) = x^2 LN(2,-:xD:) = x^2 2! L_2(-:xD:).
An example of the umbral composition in 5) is given in A038207.
The op. xDx is related to the Euler/binomial transformation for power series/o.g.f.s. through exp(t*xDx) f(x) = f[x/(1-t*x)]/(1-t*x) and to the special Moebius/linear fractional/projective transformation z exp(-t*zDz)(1/z)f(z) = f(z/(1+t*z)).
For a general discussion of umbral calculus see the Gessel link. (End)
From Wolfdieter Lang, Jan 31 2013: (Start)
Standard recurrence derived from the three term recurrence of the orthogonal polynomials system {n!*L(n,x)}: L(n,x) = (2*n  - 1 - x)*L(n-1,x) - (n-1)^2*L(n-2,x), n>=1, L(-1,x) = 0, L(0,x) = 1.
  a(n,m) = (2*n-1)*a(n-1,m) - a(n-1,m-1) - (n-1)^2*a(n-2,m),
  n >=1, with a(n,-1) = 0, a(0,0) = 1, a(n,m) = 0 if n < m. (compare this with Peter Luschny's program for the unsigned case |a(n,m)| = (-1)^m*a(n,m)).
Simplified recurrence (using column recurrence from explicit form for a(n,m) given above):
a(n,m) = (n+m)*a(n-1,m) - a(n-1,m-1), n >= 1, a(0,0) = 1, a(n,-1) = 0, a(n,m) = 0 if n < m. (End)
|T(n,k)| = [x^k] (-1)^n*U(-n,1,-x), where U(a,b,x) is Kummer's hypergeometric U function. - Peter Luschny, Apr 11 2015
T(n,k) = (-1)^k*n!*S(n,k) where S(n,k) is recursively defined by: "if k = 0 then 1 else if k > n then 0 else S(n-1,k-1)/k + S(n-1,k)". - Peter Luschny, Jun 21 2017
The unsigned case is the exponential Riordan square (see A321620) of the factorial numbers. - Peter Luschny, Dec 06 2018
Omitting the diagonal and signs, this array is generated by the commutator [D^n,x^n] = D^n x^n - x^n D^n = Sum_{i=0..n-1} ((n!/i!)^2/(n-i)!) x^i D^i on p. 9 of both papers by Belov-Kanel and Kontsevich. - Tom Copeland, Jan 23 2020

Name changed and table given by Wolfdieter Lang, Nov 28 2011

cp's OEIS Frontend

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

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A002720 Number of partial permutations of an n-set; number of n X n binary matrices with at most one 1 in each row and column.

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A007489 a(n) = Sum_{k=1..n} k!.

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A033312 a(n) = n! - 1.

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A163931 Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1.

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