cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A000166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

Original entry on oeis.org

1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, 44750731559645106, 895014631192902121, 18795307255050944540, 413496759611120779881, 9510425471055777937262
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Euler (1809) not only gives the first ten or so terms of the sequence, he also proves both recurrences a(n) = (n-1)*(a(n-1) + a(n-2)) and a(n) = n*a(n-1) + (-1)^n.
a(n) is the permanent of the matrix with 0 on the diagonal and 1 elsewhere. - Yuval Dekel, Nov 01 2003
a(n) is the number of desarrangements of length n. A desarrangement of length n is a permutation p of {1,2,...,n} for which the smallest of all the ascents of p (taken to be n if there are no ascents) is even. Example: a(3) = 2 because we have 213 and 312 (smallest ascents at i = 2). See the J. Désarménien link and the Bona reference (p. 118). - Emeric Deutsch, Dec 28 2007
a(n) is the number of deco polyominoes of height n and having in the last column an even number of cells. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. - Emeric Deutsch, Dec 28 2007
Attributed to Nicholas Bernoulli in connection with a probability problem that he presented. See Problem #15, p. 494, in "History of Mathematics" by David M. Burton, 6th edition. - Mohammad K. Azarian, Feb 25 2008
a(n) is the number of permutations p of {1,2,...,n} with p(1)!=1 and having no right-to-left minima in consecutive positions. Example a(3) = 2 because we have 231 and 321. - Emeric Deutsch, Mar 12 2008
a(n) is the number of permutations p of {1,2,...,n} with p(n)! = n and having no left to right maxima in consecutive positions. Example a(3) = 2 because we have 312 and 321. - Emeric Deutsch, Mar 12 2008
Number of wedged (n-1)-spheres in the homotopy type of the Boolean complex of the complete graph K_n. - Bridget Tenner, Jun 04 2008
The only prime number in the sequence is 2. - Howard Berman (howard_berman(AT)hotmail.com), Nov 08 2008
From Emeric Deutsch, Apr 02 2009: (Start)
a(n) is the number of permutations of {1,2,...,n} having exactly one small ascent. A small ascent in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. (Example: a(3) = 2 because we have 312 and 231; see the Charalambides reference, pp. 176-180.) [See also David, Kendall and Barton, p. 263. - N. J. A. Sloane, Apr 11 2014]
a(n) is the number of permutations of {1,2,...,n} having exactly one small descent. A small descent in a permutation (p_1,p_2,...,p_n) is a position i such that p_i - p_{i+1} = 1. (Example: a(3)=2 because we have 132 and 213.) (End)
For n > 2, a(n) + a(n-1) = A000255(n-1); where A000255 = (1, 1, 3, 11, 53, ...). - Gary W. Adamson, Apr 16 2009
Connection to A002469 (game of mousetrap with n cards): A002469(n) = (n-2)*A000255(n-1) + A000166(n). (Cf. triangle A159610.) - Gary W. Adamson, Apr 17 2009
From Emeric Deutsch, Jul 18 2009: (Start)
a(n) is the sum of the values of the largest fixed points of all non-derangements of length n-1. Example: a(4)=9 because the non-derangements of length 3 are 123, 132, 213, and 321, having largest fixed points 3, 1, 3, and 2, respectively.
a(n) is the number of non-derangements of length n+1 for which the difference between the largest and smallest fixed point is 2. Example: a(3) = 2 because we have 1'43'2 and 32'14'; a(4) = 9 because we have 1'23'54, 1'43'52, 1'53'24, 52'34'1, 52'14'3, 32'54'1, 213'45', 243'15', and 413'25' (the extreme fixed points are marked).
(End)
a(n), n >= 1, is also the number of unordered necklaces with n beads, labeled differently from 1 to n, where each necklace has >= 2 beads. This produces the M2 multinomial formula involving partitions without part 1 given below. Because M2(p) counts the permutations with cycle structure given by partition p, this formula gives the number of permutations without fixed points (no 1-cycles), i.e., the derangements, hence the subfactorials with their recurrence relation and inputs. Each necklace with no beads is assumed to contribute a factor 1 in the counting, hence a(0)=1. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 01 2010
From Emeric Deutsch, Sep 06 2010: (Start)
a(n) is the number of permutations of {1,2,...,n, n+1} starting with 1 and having no successions. A succession in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. Example: a(3)=2 because we have 1324 and 1432.
a(n) is the number of permutations of {1,2,...,n} that do not start with 1 and have no successions. A succession in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. Example: a(3)=2 because we have 213 and 321.
(End)
Increasing colored 1-2 trees with choice of two colors for the rightmost branch of nonleave except on the leftmost path, there is no vertex of outdegree one on the leftmost path. - Wenjin Woan, May 23 2011
a(n) is the number of zeros in n-th row of the triangle in A170942, n > 0. - Reinhard Zumkeller, Mar 29 2012
a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 2 pure options. - Raimundas Vidunas, Jan 22 2014
Convolution of sequence A135799 with the sequence generated by 1+x^2/(2*x+1). - Thomas Baruchel, Jan 08 2016
The number of interior lattice points of the subpolytope of the n-dimensional permutohedron whose vertices correspond to permutations avoiding 132 and 312. - Robert Davis, Oct 05 2016
Consider n circles of different radii, where each circle is either put inside some bigger circle or contains a smaller circle inside it (no common points are allowed). Then a(n) gives the number of such combinations. - Anton Zakharov, Oct 12 2016
If we partition the permutations of [n+1] in A000240 according to their starting digit, we will get (n+1) equinumerous classes each of size a(n), i.e., A000240(n+1) = (n+1)*a(n), hence a(n) is the size of each class of permutations of [n+1] in A000240. For example, for n = 4 we have 45 = 5*9. - Enrique Navarrete, Jan 10 2017
Call d_n1 the permutations of [n] that have the substring n1 but no substring in {12,23,...,(n-1)n}. If we partition them according to their starting digit, we will get (n-1) equinumerous classes each of size A000166(n-2) (the class starting with the digit 1 is empty since we must have the substring n1). Hence d_n1 = (n-1)*A000166(n-2) and A000166(n-2) is the size of each nonempty class in d_n1. For example, d_71 = 6*44 = 264, so there are 264 permutations in d_71 distributed in 6 nonempty classes of size A000166(5) = 44. (To get permutations in d_n1 recursively from more basic ones see the link "Forbidden Patterns" below.) - Enrique Navarrete, Jan 15 2017
Also the number of maximum matchings and minimum edge covers in the n-crown graph. - Eric W. Weisstein, Jun 14 and Dec 24 2017
The sequence a(n) taken modulo a positive integer k is periodic with exact period dividing k when k is even and dividing 2*k when k is odd. This follows from the congruence a(n+k) = (-1)^k*a(n) (mod k) for all n and k, which in turn is easily proved by induction making use of the recurrence a(n) = n*a(n-1) + (-1)^n. - Peter Bala, Nov 21 2017
a(n) is the number of distinct possible solutions for a directed, no self loop containing graph (not necessarily connected) that has n vertices, and each vertex has an in- and out-degree of exactly 1. - Patrik Holopainen, Sep 18 2018
a(n) is the dimension of the kernel of the random-to-top and random-to-random shuffling operators over a collection of n objects (in a vector space of size n!), as noticed by M. Wachs and V. Reiner. See the Reiner, Saliola and Welker reference below. - Nadia Lafreniere, Jul 18 2019
a(n) is the number of distinct permutations for a Secret Santa gift exchange with n participants. - Patrik Holopainen, Dec 30 2019
a(2*n+1) is even. More generally, a(m*n+1) is divisible by m*n, which follows from a(n+1) = n*(a(n) + a(n-1)) = n*A000255(n-1) for n >= 1. a(2*n) is odd; in fact, a(2*n) == 1 (mod 8). Other divisibility properties include a(6*n) == 1 (mod 24), a(9*n+4) == a(9*n+7) == 0 (mod 9), a(10*n) == 1 (mod 40), a(11*n+5) == 0 (mod 11) and a(13*n+8 ) == 0 (mod 13). - Peter Bala, Apr 05 2022
Conjecture: a(n) with n > 2 is a perfect power only for n = 4 with a(4) = 3^2. This has been verified for n <= 1000. - Zhi-Wei Sun, Jan 09 2025

Examples

			a(2) = 1, a(3) = 2 and a(4) = 9 since the possibilities are {BA}, {BCA, CAB} and {BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB, DCBA}. - _Henry Bottomley_, Jan 17 2001
The Boolean complex of the complete graph K_4 is homotopy equivalent to the wedge of 9 3-spheres.
Necklace problem for n = 6: partitions without part 1 and M2 numbers for n = 6: there are A002865(6) = 4 such partitions, namely (6), (2,4), (3^2) and (2^3) in A-St order with the M2 numbers 5!, 90, 40 and 15, respectively, adding up to 265 = a(6). This corresponds to 1 necklace with 6 beads, two necklaces with 2 and 4 beads respectively, two necklaces with 3 beads each and three necklaces with 2 beads each. - _Wolfdieter Lang_, Jun 01 2010
G.f. = 1 + x^2 + 9*x^3 + 44*x^4 + 265*x^5 + 1854*x^6 + 14833*x^7 + 133496*x^8 + ...

References

  • U. Abel, Some new identities for derangement numbers, Fib. Q., 56:4 (2018), 313-318.
  • M. Bona, Combinatorics of Permutations, Chapman & Hall/CRC, Boca Raton, Florida, 2004.
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 32.
  • R. A. Brualdi and H. J. Ryser: Combinatorial Matrix Theory, 1992, Section 7.2, p. 202.
  • Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 182.
  • Florence Nightingale David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 168.
  • Florence Nightingale David, Maurice George Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263, Table 7.5.1, row 1.
  • P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29.
  • J. M. de Saint-Martin, "Le problème des rencontres" in Quadrature, No. 61, pp. 14-19, 2006, EDP-Sciences Les Ulis (France).
  • H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 19.
  • Leonhard Euler, Solution quaestionis curiosae ex doctrina combinationum, Mémoires Académie sciences St. Pétersburg 3 (1809/1810), 57-64; also E738 in his Collected Works, series I, volume 7, pages 435-440.
  • J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
  • A. Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).
  • Irving Kaplansky, John Riordan, The problème des ménages. Scripta Math. 12 (1946), 113-124. See Eq(1).
  • Arnold Kaufmann, "Introduction à la combinatorique en vue des applications." Dunod, Paris, 1968. See p. 92.
  • Florian Kerschbaum and Orestis Terzidis, Filtering for Private Collaborative Benchmarking, in Emerging Trends in Information and Communication Security, Lecture Notes in Computer Science, Volume 3995/2006.
  • E. Lozansky and C. Rousseau, Winning Solutions, Springer, 1996; see p. 152.
  • P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 102.
  • M. S. Petković, "Non-attacking rooks", Famous Puzzles of Great Mathematicians, pp. 265-268, Amer. Math. Soc.(AMS), 2009.
  • V. Reiner, F. Saliola, and V. Welker. Spectra of Symmetrized Shuffling Operators, Memoirs of the American Mathematical Society, vol. 228, Amer. Math. Soc., Providence, RI, 2014, pp. 1-121. See section VI.9.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
  • H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 23.
  • T. Simpson, Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108.
  • 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).
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 122.
  • D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 82.
  • H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 147, Eq. 5.2.9 (q=1).

Links

Crossrefs

Cf. A000142, A002467, A003221, A000522, A000240, A000387, A000449, A000475, A129135, A092582, A000255, A002469, A159610, A068985, A068996, A047865, A038205, A008279, A281682.
For the probabilities a(n)/n!, see A053557/A053556 and A103816/A053556.
A diagonal of A008291 and A068106. Column A008290(n,0).
A001120 has a similar recurrence.
For other derangement numbers see also A053871, A033030, A088991, A088992.
Pairwise sums of A002741 and A000757. Differences of A001277.
Cf. A101560, A101559, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.
A diagonal in triangles A008305 and A010027.
a(n)/n! = A053557/A053556 = (N(n, n) of A103361)/(D(n, n) of A103360).
Column k=0 of A086764 and of A334715. Column k=1 of A364068.
Row sums of A216963 and of A323671.

Programs

  • Haskell
    a000166 n = a000166_list !! n
a000166_list = 1 : 0 : zipWith (*) [1..]
                       (zipWith (+) a000166_list $ tail a000166_list)
-- Reinhard Zumkeller, Dec 09 2012
  • Magma
    I:=[0,1]; [1] cat [n le 2 select I[n] else (n-1)*(Self(n-1)+Self(n-2)): n in [1..30]]; // Vincenzo Librandi, Jan 07 2016
  • Maple
    A000166 := proc(n) option remember; if n<=1 then 1-n else (n-1)*(procname(n-1)+procname(n-2)); fi; end;
a:=n->n!*sum((-1)^k/k!, k=0..n): seq(a(n), n=0..21); # Zerinvary Lajos, May 17 2007
ZL1:=[S,{S=Set(Cycle(Z,card>1))},labeled]: seq(count(ZL1,size=n),n=0..21); # Zerinvary Lajos, Sep 26 2007
with (combstruct):a:=proc(m) [ZL,{ZL=Set(Cycle(Z,card>=m))},labeled]; end: A000166:=a(2):seq(count(A000166,size=n),n=0..21); # Zerinvary Lajos, Oct 02 2007
Z := (x, m)->m!^2*sum(x^j/((m-j)!^2), j=0..m): R := (x, n, m)->Z(x, m)^n: f := (t, n, m)->sum(coeff(R(x, n, m), x, j)*(t-1)^j*(n*m-j)!, j=0..n*m): seq(f(0, n, 1), n=0..21); # Zerinvary Lajos, Jan 22 2008
a:=proc(n) if `mod`(n,2)=1 then sum(2*k*factorial(n)/factorial(2*k+1), k=1.. floor((1/2)*n)) else 1+sum(2*k*factorial(n)/factorial(2*k+1), k=1..floor((1/2)*n)-1) end if end proc: seq(a(n),n=0..20); # Emeric Deutsch, Feb 23 2008
G(x):=2*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]/2,n=0..21); # Zerinvary Lajos, Apr 03 2009
seq(simplify(KummerU(-n, -n, -1)), n = 0..23); # Peter Luschny, May 10 2022
  • Mathematica
    a[0] = 1; a[n_] := n*a[n - 1] + (-1)^n; a /@ Range[0, 21] (* Robert G. Wilson v *)
a[0] = 1; a[1] = 0; a[n_] := Round[n!/E] /; n >= 1 (* Michael Taktikos, May 26 2006 *)
Range[0, 20]! CoefficientList[ Series[ Exp[ -x]/(1 - x), {x, 0, 20}], x]
dr[{n_,a1_,a2_}]:={n+1,a2,n(a1+a2)}; Transpose[NestList[dr,{0,0,1},30]][[3]] (* Harvey P. Dale, Feb 23 2013 *)
a[n_] := (-1)^n HypergeometricPFQ[{- n, 1}, {}, 1]; (* Michael Somos, Jun 01 2013 *)
a[n_] := n! SeriesCoefficient[Exp[-x] /(1 - x), {x, 0, n}]; (* Michael Somos, Jun 01 2013 *)
Table[Subfactorial[n], {n, 0, 21}] (* Jean-François Alcover, Jan 10 2014 *)
RecurrenceTable[{a[n] == n*a[n - 1] + (-1)^n, a[0] == 1}, a, {n, 0, 23}] (* Ray Chandler, Jul 30 2015 *)
Subfactorial[Range[0, 20]] (* Eric W. Weisstein, Dec 31 2017 *)
nxt[{n_,a_}]:={n+1,a(n+1)+(-1)^(n+1)}; NestList[nxt,{0,1},25][[All,2]] (* Harvey P. Dale, Jun 01 2019 *)
  • Maxima
    s[0]:1$
s[n]:=n*s[n-1]+(-1)^n$
makelist(s[n],n,0,12); /* Emanuele Munarini, Mar 01 2011 */
  • PARI
    {a(n) = if( n<1, 1, n * a(n-1) + (-1)^n)}; /* Michael Somos, Mar 24 2003 */
  • PARI
    {a(n) = n! * polcoeff( exp(-x + x * O(x^n)) / (1 - x), n)}; /* Michael Somos, Mar 24 2003 */
  • PARI
    {a(n)=polcoeff(sum(m=0,n,m^m*x^m/(1+(m+1)*x+x*O(x^n))^(m+1)),n)} /* Paul D. Hanna */
  • PARI
    A000166=n->n!*sum(k=0,n,(-1)^k/k!) \\ M. F. Hasler, Jan 26 2012
  • PARI
    a(n)=if(n,round(n!/exp(1)),1) \\ Charles R Greathouse IV, Jun 17 2012
  • PARI
    apply( {A000166(n)=n!\/exp(n>0)}, [0..22]) \\ M. F. Hasler, Nov 09 2024
  • Python
    See Hobson link.
  • Python
    A000166_list, m, x = [], 1, 1
for n in range(10*2):
    x, m = x*n + m, -m
    A000166_list.append(x) # Chai Wah Wu, Nov 03 2014

Formula

(A000166 + A000522)/2 = A009179, (A000166 - A000522)/2 = A009628.
a(n) = A008290(n,0).
a(n) + A003048(n+1) = 2*n!. - D. G. Rogers, Aug 26 2006
a(n) = {(n-1)!/exp(1)}, n > 1, where {x} is the nearest integer function. - Simon Plouffe, March 1993 [This uses offset 1, see below for the version with offset 0. - Charles R Greathouse IV, Jan 25 2012]
a(0) = 1, a(n) = round(n!/e) = floor(n!/e + 1/2) for n > 0.
a(n) = n!*Sum_{k=0..n} (-1)^k/k!.
D-finite with recurrence a(n) = (n-1)*(a(n-1) + a(n-2)), n > 0.
a(n) = n*a(n-1) + (-1)^n.
E.g.f.: exp(-x)/(1-x).
a(n) = Sum_{k=0..n} binomial(n, k)*(-1)^(n-k)*k! = Sum_{k=0..n} (-1)^(n-k)*n!/(n-k)!. - Paul Barry, Aug 26 2004
The e.g.f. y(x) satisfies y' = x*y/(1-x).
Inverse binomial transform of A000142. - Ross La Haye, Sep 21 2004
In Maple notation, representation as n-th moment of a positive function on [-1, infinity]: a(n)= int( x^n*exp(-x-1), x=-1..infinity ), n=0, 1... . a(n) is the Hamburger moment of the function exp(-1-x)*Heaviside(x+1). - Karol A. Penson, Jan 21 2005
a(n) = A001120(n) - n!. - Philippe Deléham, Sep 04 2005
a(n) = Integral_{x=0..oo} (x-1)^n*exp(-x) dx. - Gerald McGarvey, Oct 14 2006
a(n) = Sum_{k=2,4,...} T(n,k), where T(n,k) = A092582(n,k) = k*n!/(k+1)! for 1 <= k < n and T(n,n)=1. - Emeric Deutsch, Feb 23 2008
a(n) = n!/e + (-1)^n*(1/(n+2 - 1/(n+3 - 2/(n+4 - 3/(n+5 - ...))))). Asymptotic result (Ramanujan): (-1)^n*(a(n) - n!/e) ~ 1/n - 2/n^2 + 5/n^3 - 15/n^4 + ..., where the sequence [1,2,5,15,...] is the sequence of Bell numbers A000110. - Peter Bala, Jul 14 2008
From William Vaughn (wvaughn(AT)cvs.rochester.edu), Apr 13 2009: (Start)
a(n) = Integral_{p=0..1} (log(1/(1-p)) - 1)^n dp.
Proof: Using the substitutions 1=log(e) and y = e(1-p) the above integral can be converted to ((-1)^n/e) Integral_{y=0..e} (log(y))^n dy.
From CRC Integral tables we find the antiderivative of (log(y))^n is (-1)^n n! Sum_{k=0..n} (-1)^k y(log(y))^k / k!.
Using the fact that e(log(e))^r = e for any r >= 0 and 0(log(0))^r = 0 for any r >= 0 the integral becomes n! * Sum_{k=0..n} (-1)^k / k!, which is line 9 of the Formula section. (End)
a(n) = exp(-1)*Gamma(n+1,-1) (incomplete Gamma function). - Mark van Hoeij, Nov 11 2009
G.f.: 1/(1-x^2/(1-2x-4x^2/(1-4x-9x^2/(1-6x-16x^2/(1-8x-25x^2/(1-... (continued fraction). - Paul Barry, Nov 27 2009
a(n) = Sum_{p in Pano1(n)} M2(p), n >= 1, with Pano1(n) the set of partitions without part 1, and the multinomial M2 numbers. See the characteristic array for partitions without part 1 given by A145573 in Abramowitz-Stegun (A-S) order, with A002865(n) the total number of such partitions. The M2 numbers are given for each partition in A-St order by the array A036039. - Wolfdieter Lang, Jun 01 2010
a(n) = row sum of A008306(n), n > 1. - Gary Detlefs, Jul 14 2010
a(n) = ((-1)^n)*(n-1)*hypergeom([-n+2, 2], [], 1), n>=1; 1 for n=0. - Wolfdieter Lang, Aug 16 2010
a(n) = (-1)^n * hypergeom([ -n, 1], [], 1), n>=1; 1 for n=0. From the binomial convolution due to the e.g.f. - Wolfdieter Lang, Aug 26 2010
Integral_{x=0..1} x^n*exp(x) = (-1)^n*(a(n)*e - n!).
O.g.f.: Sum_{n>=0} n^n*x^n/(1 + (n+1)*x)^(n+1). - Paul D. Hanna, Oct 06 2011
Abs((a(n) + a(n-1))*e - (A000142(n) + A000142(n-1))) < 2/n. - Seiichi Kirikami, Oct 17 2011
G.f.: hypergeom([1,1],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011
From Sergei N. Gladkovskii, Nov 25 2011, Jul 05 2012, Sep 23 2012, Oct 13 2012, Mar 09 2013, Mar 10 2013, Oct 18 2013: (Start)
Continued fractions:
In general, e.g.f. (1+a*x)/exp(b*x) = U(0) with U(k) = 1 + a*x/(1-b/(b-a*(k+1)/U(k+1))). For a=-1, b=-1: exp(-x)/(1-x) = 1/U(0).
E.g.f.: (1-x/(U(0)+x))/(1-x), where U(k) = k+1 - x + (k+1)*x/U(k+1).
E.g.f.: 1/Q(0) where Q(k) = 1 - x/(1 - 1/(1 - (k+1)/Q(k+1))).
G.f.: 1/U(0) where U(k) = 1 + x - x*(k+1)/(1 - x*(k+1)/U(k+1)).
G.f.: Q(0)/(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))).
G.f.: 1/Q(0) where Q(k) = 1 - 2*k*x - x^2*(k + 1)^2/Q(k+1).
G.f.: T(0) where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2-(1-2*x*k)*(1-2*x-2*x*k)/T(k+1)). (End)
0 = a(n)*(a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*a(n+2) if n>=0. - Michael Somos, Jan 25 2014
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(k + x)^k*(k + x + 1)^(n-k) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(k + x)^(n-k)*(k + x - 1)^k, for arbitrary x. - Peter Bala, Feb 19 2017
From Peter Luschny, Jun 20 2017: (Start)
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(-j-1, -n-1)*abs(Stirling1(j, k)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Pochhammer(n-k+1, k) (cf. A008279). (End)
a(n) = n! - Sum_{j=0..n-1} binomial(n,j) * a(j). - Alois P. Heinz, Jan 23 2019
Sum_{n>=2} 1/a(n) = A281682. - Amiram Eldar, Nov 09 2020
a(n) = KummerU(-n, -n, -1). - Peter Luschny, May 10 2022
a(n) = (-1)^n*Sum_{k=0..n} Bell(k)*Stirling1(n+1, k+1). - Mélika Tebni, Jul 05 2022

Extensions

Minor edits by M. F. Hasler, Jan 16 2017

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

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

References

  • 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).

Links

Crossrefs

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.

Programs

  • Magma
    [1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // Arkadiusz Wesolowski, May 17 2014
  • Maple
    seq(mul(k, k=3..n), n=0..20); # Zerinvary Lajos, Sep 14 2007
  • Mathematica
    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)};
  • PARI
    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
  • PARI
    A001710=n->n!\2+(n<2) \\ M. F. Hasler, Dec 01 2013
  • Python
    from math import factorial
def A001710(n): return factorial(n)>>1 if n > 1 else 1 # Chai Wah Wu, Feb 14 2023
  • SageMath
    def A001710(n): return (factorial(n) +int(n<2))//2
[A001710(n) for n in range(31)] # G. C. Greubel, Sep 28 2024
  • Scheme
    ;; 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

Formula

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)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001
Further terms from Simone Severini, Oct 15 2004

A002627 a(n) = n*a(n-1) + 1, a(0) = 0.

Original entry on oeis.org

0, 1, 3, 10, 41, 206, 1237, 8660, 69281, 623530, 6235301, 68588312, 823059745, 10699776686, 149796873605, 2246953104076, 35951249665217, 611171244308690, 11001082397556421, 209020565553572000, 4180411311071440001, 87788637532500240022
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence shares divisibility properties with A000522; each of the primes in A072456 divide only a finite number of terms of this sequence. - T. D. Noe, Jul 07 2005
Sum of the lengths of the first runs in all permutations of [n]. Example: a(3)=10 because the lengths of the first runs in the permutation (123),(13)2,(3)12,(2)13,(23)1 and (3)21 are 3,2,1,1,2 and 1, respectively (first runs are enclosed between parentheses). Number of cells in the last columns of 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. a(n) = Sum_{k=1..n} k*A092582(n,k). - Emeric Deutsch, Aug 16 2006
Starting with offset 1 = eigensequence of an infinite lower triangular matrix with (1, 2, 3, ...) as the right border, (1, 1, 1, ...) as the left border, and the rest zeros. - Gary W. Adamson, Apr 27 2009
Sums of rows of the triangle in A173333, n > 0. - Reinhard Zumkeller, Feb 19 2010
if s(n) is a sequence defined as s(0) = x, s(n) = n*s(n-1)+k, n > 0 then s(n) = n!*x + a(n)*k. - Gary Detlefs, Feb 20 2010
Number of arrangements of proper subsets of n distinct objects, i.e., arrangements which are not permutations (where the empty set is considered a proper subset of any nonempty set); see example. - Daniel Forgues, Apr 23 2011
For n >= 0, A002627(n+1) is the sequence of sums of Pascal-like triangle with one side 1,1,..., and the other side A000522. - Vladimir Shevelev, Feb 06 2012
a(n) = q(n,1) for n >= 1, where the polynomials q are defined at A248669. - Clark Kimberling, Oct 11 2014
a(n) is the number of quasilinear weak orderings on {1,...,n}. - J. Devillet, Dec 22 2017

Examples

			[a(0), a(1), ...] = GAMMA(m+1,1)*exp(1) - GAMMA(m+1) = [exp(-1)*exp(1)-1, 2*exp(-1)*exp(1)-1, 5*exp(-1)*exp(1)-2, 16*exp(-1)*exp(1)-6, 65*exp(-1)*exp(1)-24, 326*exp(-1)*exp(1)-120, ...]. - _Stephen Crowley_, Jul 24 2009
From _Daniel Forgues_, Apr 25 2011: (Start)
  n=0: {}: #{} = 0
  n=1: {1}: #{()} = 1
  n=2: {1,2}: #{(),(1),(2)} = 3
  n=3: {1,2,3}: #{(),(1),(2),(3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)} = 10
(End)
x + 3*x^2 + 10*x^3 + 41*x^4 + 206*x^5 + 1237*x^6 + 8660*x^7 + 69281*x^8 + ...

References

  • 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).

Links

Crossrefs

Cf. A000522, A001113, A092582.
Second diagonal of A059922, cf. A056542.
Conjectured to give records in A130147.

Programs

  • Haskell
    a002627 n = a002627_list !! n
a002627_list = 0 : map (+ 1) (zipWith (*) [1..] a002627_list)
-- Reinhard Zumkeller, Mar 24 2013
  • Magma
    I:=[1]; [0] cat [n le 1 select I[n] else n*Self(n-1)+1:n in [1..21]]; // Marius A. Burtea, Aug 07 2019
  • Maple
    A002627 := proc(n)
    add( (n-j)!*binomial(n,j), j=1..n) ;
end proc:
seq(A002627(n),n=0..21) ; # Zerinvary Lajos, Jul 31 2006
  • Mathematica
    FoldList[ #1*#2 + 1 &, 0, Range[21]] (* Robert G. Wilson v, Oct 11 2005 *)
RecurrenceTable[{a[0]==0,a[n]==n*a[n-1]+1},a,{n,30}] (* Harvey P. Dale, Mar 29 2015 *)
  • Maxima
    makelist(sum(n!/k!,k,1,n),n,0,40); /* Emanuele Munarini, Jun 20 2014 */
  • PARI
    a(n)= n!*sum(k=1,n, 1/k!); \\ Joerg Arndt, Apr 24 2011

Formula

a(n) = n! * Sum_{k=1..n} 1/k!.
a(n) = A000522(n) - n!. - Michael Somos, Mar 26 1999
a(n) = floor( n! * (e-1) ), n >= 1. - Amarnath Murthy, Mar 08 2002
E.g.f.: (exp(x)-1)/(1-x). - Mario Catalani (mario.catalani(AT)unito.it), Feb 06 2003
Binomial transform of A002467. - Ross La Haye, Sep 21 2004
a(n) = Sum_{j=1..n} (n-j)!*binomial(n,j). - Zerinvary Lajos, Jul 31 2006
a(n) = 1 + Sum_{k=0..n-1} k*a(k). - Benoit Cloitre, Jul 26 2008
a(m) = Integral_{s=0..oo} ((1+s)^m - s^m)*exp(-s) = GAMMA(m+1,1) * exp(1) - GAMMA(m+1). - Stephen Crowley, Jul 24 2009
From Sergei N. Gladkovskii, Jul 05 2012: (Start)
a(n+1) = A000522(n) + A001339(n) - A000142(n+1);
E.g.f.: Q(0)/(1-x), where Q(k)= 1 + (x-1)*k!/(1 - x/(x + (x-1)*(k+1)!/Q(k+1))); (continued fraction). (End)
E.g.f.: x/(1-x)*E(0)/2, where E(k)= 1 + 1/(1 - x/(x + (k+2)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
1/(e - 1) = 1 - 1!/(1*3) - 2!/(3*10) - 3!/(10*41) - 4!/(41*206) - ... (see A056542 and A185108). - Peter Bala, Oct 09 2013
Conjecture: a(n) + (-n-1)*a(n-1) + (n-1)*a(n-2) = 0. - R. J. Mathar, Feb 16 2014
The e.g.f. f(x) = (exp(x)-1)/(1-x) satisfies the differential equation: (1-x)*f'(x) - (2-x)*f(x) + 1, from which we can obtain the recurrence:
a(n+1) = a(n) + n! + Sum_{k=1..n} (n!/k!)*a(k). The above conjectured recurrence can be obtained from the original recurrence or from the differential equation satisfied by f(x). - Emanuele Munarini, Jun 20 2014
Limit_{n -> oo} a(n)/n! = exp(1) - 1. - Carmine Suriano, Jul 01 2015
Product_{n>=2} a(n)/(a(n)-1) = exp(1) - 1. See A091131. - James R. Buddenhagen, Jul 21 2019
a(n) = Sum_{k=0..n-1} k!*binomial(n,k). - Ridouane Oudra, Jun 17 2025

Extensions

Comments from Michael Somos

A092956 a(n) = (2*n+2)!/((n+2)*n!).

Original entry on oeis.org

1, 8, 90, 1344, 25200, 570240, 15135120, 461260800, 15878903040, 609493248000, 25812039052800, 1195656969830400, 60138698780160000, 3264143527636992000, 190165504623494400000, 11836497605427855360000, 783921372659482337280000
Offset: 0

Views

Author

Amarnath Murthy, Mar 25 2004

Keywords

Links

Crossrefs

Row sums of A105725.
Cf. A092582.

Programs

  • Magma
    [Factorial(n+1)*Binomial(2*n+2, n): n in [0..20]]; // G. C. Greubel, Aug 11 2022
  • Maple
    seq((2*n+2)!/(n+2)/n!,n=0..17); # Emeric Deutsch
a:=n->sum(mul (j-k+n,j=1..n),k=1..n): seq(a(n),n=1..17); # Zerinvary Lajos, Jun 04 2007
  • Mathematica
    Table[(2n+2)!/((n+2) n!), {n, 0, 16}] (* Bruno Berselli, Mar 06 2013 *)
  • Maxima
    A092956(n):=(2*n+2)!/((n+2)*n!)$ makelist(A092956(n),n,0,30); /* Martin Ettl, Nov 05 2012 */
  • SageMath
    [factorial(n+1)*binomial(2*n+2,n) for n in (0..20)] # G. C. Greubel, Aug 11 2022

Formula

a(n) = Sum_{k=1..n+1} Gamma(n+1+k)/Gamma(k). - Bruno Berselli, Mar 06 2013
Let E(x) = Sum_{n>=0} a(n)*x^(2*n)/n!, then E(x) = 2- E(0,x), where E(k,x) = 1 - x^2*(k+1)/( x^2*(k+1) + (k + 1 -x^2)*(k + 2 -x^2)/E(k+1,x) ); (continued fraction). - Sergei N. Gladkovskii, Oct 21 2013
a(n) = A092582(2n+2, n+1). - Alois P. Heinz, Jun 19 2017
From G. C. Greubel, Aug 11 2022: (Start)
G.f.: Hypergeometric2F1([2,2,3/2], [3], 4*x).
E.g.f.: 4*x*Hypergeometric2F1([5/2,3], [4], 4*x) + Hypergeometric2F1([3/2,2], [3], 4*x). (End)

Extensions

More terms from Emeric Deutsch, Apr 18 2005
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007
More terms from Zerinvary Lajos, Jun 04 2007

A130477 T(n,k) is the number of permutations of [n] with maximum descent k, T(n,k) for n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 8, 12, 1, 4, 15, 40, 60, 1, 5, 24, 90, 240, 360, 1, 6, 35, 168, 630, 1680, 2520, 1, 7, 48, 280, 1344, 5040, 13440, 20160, 1, 8, 63, 432, 2520, 12096, 45360, 120960, 181440, 1, 9, 80, 630, 4320, 25200, 120960, 453600, 1209600, 1814400, 1, 10, 99, 880, 6930, 47520, 277200, 1330560, 4989600, 13305600, 19958400
Offset: 0

Views

Author

Gary W. Adamson, May 28 2007

Keywords

Comments

Previous name was: Triangle generated from finite differences of A130461.
Decomposition of the permutations.
From Liam Solus, Aug 23 2018: (Start)
For k>0, T(n,k) equals the number of permutations p of [n] such that the largest index i for which p(i)>p(i+1) is k; i.e., T(n,k) is the number of permutations of [n] with maximum descent being k. See Lemma 3.4 of the paper by L. Solus below.
When T(n,k) is taken as the weight of coordinate x_k for k = 0,...,n-1 in an (n-1)-dimensional weighted projective space, the result is the toric variety defined by an n-dimensional simplex whose Ehrhart h^*-polynomial is the n-th Eulerian polynomial. See Theorem 3.5 of the paper by L. Solus below.
(End)

Examples

			First few rows of the triangle A130461 = (1; 1, 1; 1, 1, 1; 1, 1, 2, 1; 1, 1, 2, 3, 1; 1, 1, 2, 6, 4, 1;...). Deleting the left border and taking finite differences at the top of each remaining column, we get the first few rows of this triangle:
1;
1, 1;
1, 2,  3;
1, 3,  8,  12;
1, 4, 15,  40,  60;
1, 5, 24,  90, 240,  360;
1, 6, 35, 168, 630, 1680, 2520;
...

Links

Crossrefs

The triangle with each row reversed is A092582.
Cf. A000142 (row sums), A001710 (main diagonal), A008292.
Cf. A130460, A130461, A130476, A130478.

Programs

  • Maple
    T := (n,k) -> (n-k+1+0^k)*((n+1)!/(n-k+2)!):
seq(seq(T(n,k),k=0..n),n=0..10); # Peter Luschny, Sep 17 2018
  • Mathematica
    Flatten[Table[Table[(n - k + 1 + 0^k)*(n + 1)!/(n - k + 2)!, {k,0,n}], {n, 0, 10}], 1] (* Olivier Gérard, Aug 04 2012 *)
  • PARI
    {T(n, k) = if( n<1 || k>n, 0, k==1, 1, n! * (n+1-k) / (n+2-k)!)}; /* Michael Somos, Jun 25 2017 */

Formula

Each term in n-th row divides n!.
Given triangle A130461 and deleting the left border (1,1,1,...) take finite differences by columns and reorient into rows.
T(n,k) = (n-k+1+0^k)*((n+1)!/(n-k+2)!) - Olivier Gérard, Aug 04 2012

Extensions

New name using a comment by Liam Solus, Peter Luschny, Sep 17 2018

A092580 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which exactly the first k terms satisfy the up-down property, i.e., p(1)p(3), p(3)

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 12, 4, 3, 5, 60, 20, 15, 9, 16, 360, 120, 90, 54, 35, 61, 2520, 840, 630, 378, 245, 155, 272, 20160, 6720, 5040, 3024, 1960, 1240, 791, 1385, 181440, 60480, 45360, 27216, 17640, 11160, 7119, 4529, 7936, 1814400, 604800, 453600, 272160, 176400, 111600, 71190, 45290, 28839, 50521
Offset: 1

Views

Author

Emeric Deutsch and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004

Keywords

Comments

Row sums are the factorial numbers (A000142). First column is A001710. Second column is A001715. Diagonal is A000111.

Examples

			T(4,3)=3 because 1432, 2431, 3421 are the only permutations of [4] in which exactly the first 3 entries satisfy the up-down property.
Triangle starts:
    1;
    1,   1;
    3,   1,  2;
   12,   4,  3,  5;
   60,  20, 15,  9, 16;
  360, 120, 90, 54, 35, 61;
  ...

Links

Crossrefs

Cf. A000111, A000142, A001710, A001715, A092582, A230960.
T(2n,n) gives A362603.

Programs

  • Maple
    b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
E:= n-> b(n, 0):
T:= (n, k)-> `if`(n=k, E(n), n!*((k+1)*E(k)-E(k+1))/(k+1)!):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 12 2016
  • Mathematica
    b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[o - 1 + j, u - j], {j, 1, u}]]; e[n_] := b[n, 0]; T[n_, k_] := If[n == k, e[n], n!*((k + 1)*e[k] - e[k + 1])/(k + 1)!]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)

Formula

T(n, k) = n!*[(k+1)*E(k)-E(k+1)]/(k+1)! for k=0} [E(n)x^n/n!] (i.e., E(n) = A000111(n)).
Sum_{k=0..n} (k+1) * T(n,k) = A230960(n). - Alois P. Heinz, Apr 27 2023

A117396 Triangle, read by rows, defined by: T(n,k) = (k+1)*T(n,k+1) - Sum_{j=1..n-k-1} T(j,0)*T(n,j+k+1) for n>k with T(n,n)=1 for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 11, 4, 1, 1, 77, 51, 19, 5, 1, 1, 437, 291, 109, 29, 6, 1, 1, 2957, 1971, 739, 197, 41, 7, 1, 1, 23117, 15411, 5779, 1541, 321, 55, 8, 1, 1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1, 1, 2018957, 1345971, 504739, 134597, 28041, 4807, 701, 89, 10, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2006

Keywords

Comments

Columns equal the partial sums of columns of triangle A092582 for k>0: T(n, k) - T(n-1, k) = A092582(n,k) = number of permutations p of [n] having length of first run equal to k.

Examples

			Triangle begins:
  1;
  1,      1;
  1,      2,      1;
  1,      5,      3,     1;
  1,     17,     11,     4,     1;
  1,     77,     51,    19,     5,    1;
  1,    437,    291,   109,    29,    6,   1;
  1,   2957,   1971,   739,   197,   41,   7,  1;
  1,  23117,  15411,  5779,  1541,  321,  55,  8, 1;
  1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1; ...
Matrix inverse is:
   1;
  -1,  1;
   1, -2,  1;
   1,  1, -3,  1;
   1,  1,  1, -4,  1;
   1,  1,  1,  1, -5, 1; ...
Matrix log is the integer triangle A117398:
    0;
    1,  0;
    0,  2,  0;
   -1,  2,  3,  0;
   -3,  4,  5,  4,  0;
   -9, 14, 15,  9,  5,  0;
  -33, 68, 65, 34, 14,  6,  0; ...

Links

Crossrefs

Cf. A014288 (column 1), A056199 (column 2), A117397 (column 3), A003422 (row sums), A117398 (matrix log); A092582.

Programs

  • Magma
    [k eq 0 select 1 else k*(&+[Factorial(j)/Factorial(k+1): j in [k-1..n]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 24 2021
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0, 1, k*Sum[j!/(k+1)!, {j,k-1,n}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 24 2021 *)
  • PARI
    T(n,k)=if(n
  • PARI
    /* Definition by Matrix Inverse: */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,if(r==c+1,-c,1))));(M^-1)[n+1,k+1]
  • PARI
    T(n,k)=if(nPaul D. Hanna, Jun 20 2006
  • Sage
    def A117396(n,k): return 1 if (k==0) else k*sum(factorial(j)/factorial(k+1) for j in (k-1..n))
flatten([[A117396(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 24 2021

Formula

T(n,k) = k*Sum_{j=k-1..n} j!/(k+1)! for n >= k > 0, with T(n,0) = 1 for n >= 0. - Paul D. Hanna, Jun 20 2006

A117397 Column 3 of triangle A117396.

Original entry on oeis.org

1, 4, 19, 109, 739, 5779, 51139, 504739, 5494339, 65369539, 843747139, 11741033539, 175200329539, 2790549065539, 47251477577539, 847548190793539, 16053185741897539, 320165936763977539, 6706533708227657539, 147206624680428617539, 3378708717041050697539
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2006

Keywords

Comments

Equals the partial sums of column 3 of triangle A092582.

Examples

			G.f.: A(x) = 1 + 4*x + 19*x^2 + 109*x^3 + 739*x^4 + 5779*x^5 + 51139*x^6 + 504739*x^7 + 5494339*x^8 + 65369539*x^9 + 843747139*x^10 + ...

Links

Crossrefs

Cf. A117396 (triangle), A014288 (column 1), A056199 (column 2), A003422 (row sums).
Cf. A003422, A007489, A092582.

Programs

  • Magma
    [(&+[Factorial(j): j in [2..n+3]])/8: n in [0..30]]; // G. C. Greubel, Sep 05 2022
  • Maple
    a:=n->sum(j!/8,j=2..n): seq(a(n), n=3..21); # Zerinvary Lajos, Jan 08 2007
  • Mathematica
    Table[Sum[i!/8, {i, 2, n}], {n, 3, 21}] (* Zerinvary Lajos, Jul 12 2009 *)
  • PARI
    {a(n)=1+sum(k=4,n+3,k!)*3/4!}
for(n=0,25,print1(a(n),", "))
  • SageMath
    [sum(factorial(j) for j in (2..n+3))/8 for n in (0..30)] # G. C. Greubel, Sep 05 2022

Formula

G.f. satisfies A(x) = (1-x)/(1 - 5*x + 5*x^2) * (1 + x^2*A'(x)).
a(n) = 1 + Sum_{k=4..n+3} k!*3/4! for n > 0, with a(0)=1.
G.f.: W(0)/(8*x*(1-x)) -1/(4*x), where W(k) = 1 + 1/( 1 - x*(k+3)/( x*(k+3) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 20 2013
G.f.: (Sum_{n>=0} (n+2)!*x^n)/(8*x*(1-x)) - 1/(4*x). - Sergei N. Gladkovskii, Aug 20 2013
a(n) = (1/8)*(A007489(n+3) - 1) = (1/8)*(A003422(n+4) - 2). - G. C. Greubel, Sep 05 2022
Showing 1-8 of 8 results.

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