|
| |
|
|
A001563
|
|
a(n) = n*n! = (n+1)! - n!.
(Formerly M3545 N1436)
|
|
76
|
|
|
|
0, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000, 48658040163532800000, 1072909785605898240000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
A similar sequence, with the initial 0 replaced by 1, namely A094258, is defined by the recurrence a(2) = 1, a(n) = a(n-1)*(n-1)^2/(n-2). - Andrey Ryshevich (ryshevich(AT)notes.idlab.net), May 21 2002
Denominators in power series expansion of E_1(x) + gamma + log(x), x>0. - Michael Somos, Dec 11 2002
If all the permutations of any length k are arranged in lexicographic order, the n-th term in this sequence (n <= k) gives the index of the permutation that rotates the last n elements one position to the right. E.g. there are 24 permutations of 4 items. In lexicographic order they are (0,1,2,3), (0,1,3,2), (0,2,1,3),... (3,2,0,1), (3,2,1,0). Permutation 0 is (0,1,2,3) which rotates the last 1 element, i.e. is makes no change. Permutation 1 is (0,1,3,2) which rotates the last 2 element. Pwermutation 4 is (0,3,1,2) which rotates the last 3 elements. Permutation 18 is (3,0,1,2) which rotates the last 4 elements. The same numbers work for permutations of any length. - Henry H. Rich (glasss(AT)bellsouth.net), Sep 27 2003
Stirling transform of a(n+1)=[4,18,96,600,...] is A083140(n+1)=[4,22,154,...]. - Michael Somos, Mar 04 2004
Stirling transform of a(n)=[1,4,18,96,...] is A06931(n)=[1,5,31,233,...]. - Michael Somos, Apr 27 2012
Partial sums of a(n)=[0,1,4,18,...] is A033312(n+1)=[0,1,5,23,...]. - Michael Somos, Apr 27 2012
Binomial transform of A000166(n+1)=[0,1,2,9,...] is a(n)=[0,1,4,18,...]. - Michael Somos, Apr 27 2012
Binomial transform of A000255(n+1)=[1,3,11,53,...] is a(n+1)=[1,4,18,96,...]. - Michael Somos, Apr 27 2012
Binomial transform of a(n)=[0,1,4,18,...] is A093964(n)=[0,1,6,33,...]. - Michael Somos, Apr 27 2012
Partial sums of A001564(n)=[1,3,4,14,...] is a(n+1)=[1,4,18,96,...]. - Michael Somos, Apr 27 2012
Number of small descents in all permutations of [n+1]. A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) =1. Example: a(2)=4 because there are 4 small descents in the permutations 123, 13\2, 2\13, 231, 312, 3\2\1 of {1,2,3} (shown by \). a(n)=Sum(k*A123513(n,k), k=0..n-1). - Emeric Deutsch, Oct 02 2006
a(n-1) is the number of permutations of n in which n is not fixed; equivalently, the number of permutations of the positive integers in which n is the largest element that is not fixed. - Franklin T. Adams-Watters, Nov 29 2006
Number of factors in a determinant when writing down all multiplication permutations. [From Mats Granvik, Sep 12 2008]
a(n) is also the sum of the positions of the left-to-right maxima in all permutations of [n]. Example: a(3)=18 because the positions of the left-to-right maxima in the permutations 123,132,213,231,312 and 321 of [3] are 123, 12, 13, 12, 1 and 1, respectively and 1+2+3+1+2+1+3+1+2+1+1=18. [From Emeric Deutsch, Sep 21 2008]
Equals eigensequence of triangle A002024 ("n appears n times". [From Gary W. Adamson, Dec 29 2008]
Contribution from Gary W. Adamson, Apr 17 2009: (Start)
Preface the series with another 1: (1, 1, 4, 18,...); then the next term =
dot product of the latter with "n occurs n times". Example: 96 = (1, 1, 4, 8)
dot (4, 4, 4, 4) = (4 + 4 + 16 + 72). (End)
Row lengths of the triangle in A030298. [Reinhard Zumkeller, Mar 29 2012]
|
|
|
REFERENCES
|
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 218.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 336.
A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.
Jolley, Summation of series, Dover (1961).
Mundfrom, Daniel J.; A problem in permutations: the game of `Mousetrap'. European J. Combin. 15 (1994), no. 6, 555-560.
I. Kortchemski, Asymptotic behavior of permutation records, arXiv: 0804.0446v2 [math.CO], 18 May 2008. [From Emeric Deutsch, Sep 21 2008]
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
|
T. D. Noe, Table of n, a(n) for n=0..100
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 30
Eric Weisstein's World of Mathematics, Exponential Integral.
|
|
|
FORMULA
|
E.g.f.: x / (1 - x)^2. a(n) = -A021009(n, 1), n >= 0.- Michael Somos, Dec 11 2002
The coefficient of y^(n-1) in expansion of (y+n!)^n, n >= 1, gives the sequence 1, 4, 18, 96, 600, 4320, 35280, ... - Artur Jasinski, Oct 22 2007
Integral representation as n-th moment of a function on a positive half-axis, in Maple notation: a(n)=int(x^n*(x*(x-1)*exp(-x)), x=0..infinity), n=0, 1... This representation may not be unique. - Karol A. Penson, Sep 27 2001
a(0)=0, a(n)=n*a(n-1)+n! - Benoit Cloitre, Feb 16 2003
a(0) = 0, a(n) = (n - 1) * (1 + Sum i=1..n-1 a(i)) for i > 0 - Gerald McGarvey, Jun 11 2004
Arises in the denominators of the following identities: Sum_{1..oo}1/(n(n+1)(n+2)) = 1/4, Sum_{1..oo}1/(n(n+1)(n+2)(n+3)) = 1/18, Sum_{1..oo}1/(n(n+1)(n+2)(n+3)(n+4)) = 1/96, etc. The general expression is Sum_{n = k..infinity } 1/C(n, k) = k/(k-1). - Dick Boland, Jun 06 2005
a(n)= sum(|Stirling1(n+1, m)|, m=2..n+1), n>=1 and a(0):=0, where Stirling1(n, m)= A048994(n, m), n>=>m=0.
a(n) = 1/sum(k!/(n+k+1)!,k=0..infinity), n>0. - Vladeta Jovovic, Sep 13 2006
a(n)=Sum(k*A143946(n,k),k=1..n(n+1)/2). [From Emeric Deutsch, Sep 21 2008]
The reciprocals of a(n) are the lead coefficients in the factored form of the polynomials obtained by summing the binomial coefficients with a fixed lower term up to n as the upper term, divided by the term index, for n >= 1: sum_{k = i..n} C(k, i)/k = (1/a(n))*n*(n-1)*..*(n-i+1). The first few such polynomials are: sum_{k = 1..n} C(k, 1)/k = (1/1)*n, sum_{k = 2..n} C(k, 2)/k = (1/4)*n*(n-1), sum_{k = 3..n} C(k, 3)/k = (1/18)*n*(n-1)*(n-2), sum_{k = 4..n} C(k, 4)/k = (1/96)*n*(n-1)*(n-2)*(n-3) etc. [From Peter Breznay (breznayp(AT)uwgb.edu), Sep 28 2008]
If we define f(n,i,x)= sum(sum(binomial(k,j)*stirling1(n,k)*stirling2(j,i)*x^(k-j),j=i..k),k=i..n) then a(n)=(-1)^(n-1)*f(n,1,-2), (n>=1). [From Milan Janjic, Mar 01 2009]
Contribution from Johannes W. Meijer, Oct 16 2009: (Start)
a(0) = 0, a(1) = 1 and a(n) = (1+(n-1))^2*product((1+k), k=1..n-2), n=>2.
(End)
Sum_{n=1..infinity} (-1)^(n+1)/a(n) = 0.796599599... [Jolley eq. 289]
G.f.: 2*x*Q(0), where Q(k)= 1 - 1/(k+2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3)-1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
|
|
|
EXAMPLE
|
E_1(x) + gamma + log(x) = x/1 - x^2/4 + x^3/18 - x^4/96 + ..., x>0. - Michael Somos, Dec 11 2002
x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 4320*x^6 + 35280*x^7 + 322560*x^8 + ...
|
|
|
MAPLE
|
A001563 := n->n*n!;
a:=n->sum(sum(stirling1(n, k), j=2..n), k=2..n): seq(abs(a(n)), n=1..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007
spec := [S, {S = Union(Prod(Union(Z, Z, Z), Sequence(Z), Sequence(Z)), Prod(Union(Z, Z), Sequence(Z), Sequence(Z)))}, labelled]: seq(combstruct[count](spec, size=n)/5, n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2008
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(1):seq(count(ZLL, size=n)*n, n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 11 2008
restart: G(x):=x/(1-x)^2: f[0]:=G(x): for n from 1 to 24 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..21); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 01 2009]
restart:printlevel := -1; a := [0]; T := x->LambertW(-x); f := series((T(x)*(1+T(x)))/(1-T(x)), x, 24); for m from 1 to 21 do a := [op(a), op(2*m, f)*m! ] od; print(a); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2009]
|
|
|
MATHEMATICA
|
a = {}; Do[k = CoefficientList[Expand[(y + n!)^n], y]; AppendTo[a, k[[Length[k] - 1]]], {n, 1, 50}]; a - Artur Jasinski, Oct 22 2007
..and/or..a=s=1; lst={}; Do[a=s*n-s; s+=a; AppendTo[lst, a], {n, 2*4!}]; lst [From Vladimir Joseph Stephan Orlovsky, Jun 27 2009]
Table[Sum[n!, {i, 1, n}], {n, 0, 21}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
Table[(-1)^(n + 1)* Sum[(-1)^(n - k) k (-1)^(n - k) StirlingS1[n + 2, k + 2], {k, 0, n}], {n, 0, 21}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]
Table[n!n, {n, 0, 25}] (* From Harvey P. Dale, Oct 03 2011 *)
|
|
|
PROG
|
(PARI) {a(n) = if( n<0, 0, n * n!)} /* Michael Somos, Dec 11 2002 */
|
|
|
CROSSREFS
|
Cf. A047920, A047922, A000142, A055089, A053495.
Cf. A123513.
A143946 [From Emeric Deutsch, Sep 21 2008]
A002024 [From Gary W. Adamson, Dec 29 2008]
Contribution from Johannes W. Meijer, Oct 16 2009: (Start)
Cf. A163931 (E(x,m,n)), A002775 (n^2*n!), A091363 (n^3*n!), A091364 (n^4*n!).
(End)
Sequence in context: A180567 A152392 * A094304 A094258 A086681 A054139
Adjacent sequences: A001560 A001561 A001562 * A001564 A001565 A001566
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
