%I A001563 M3545 N1436
%S A001563 0,1,4,18,96,600,4320,35280,322560,3265920,36288000,439084800,5748019200,
%T A001563 80951270400,1220496076800,19615115520000,334764638208000,6046686277632000,
%U A001563 115242726703104000,2311256907767808000,48658040163532800000,1072909785605898240000
%N A001563 a(n) = n*n! = (n+1)!-n!.
%C A001563 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
%C A001563 Denominators in power series expansion of E_1(x)+gamma+log(x), n>0.
%C A001563 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
%C A001563 Stirling transform of a(n+1)=[4,18,96,600,...] is A083140(n+1)=[4,22,
               154,...]. - Michael Somos Mar 04 2004
%C A001563 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 (deutsch(AT)duke.poly.edu), 
               Oct 02 2006
%C A001563 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 
               (FrankTAW(AT)Netscape.net), Nov 29 2006
%C A001563 Number of factors in a determinant when writing down all multiplication 
               permutations. [From Mats O. Granvik (mgranvik(AT)abo.fi), Sep 12 
               2008]
%C A001563 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 (deutsch(AT)duke.poly.edu), Sep 21 2008]
%C A001563 Equals eigensequence of triangle A002024 ("n appears n times". [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2008]
%C A001563 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2009: 
               (Start)
%C A001563 Preface the series with another 1: (1, 1, 4, 18,...); then the next term 
               =
%C A001563 dot product of the latter with "n occurs n times". Example: 96 = (1, 
               1, 4, 8)
%C A001563 dot (4, 4, 4, 4) = (4 + 4 + 16 + 72). (End)
%D A001563 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001563 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001563 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 218.
%D A001563 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 336.
%D A001563 A. van Heemert, Cyclic permutations with sequences and related problems, 
               J. Reine Angew. Math., 198 (1957), 56-72.
%D A001563 Mundfrom, Daniel J.; A problem in permutations: the game of `Mousetrap'. 
               European J. Combin. 15 (1994), no. 6, 555-560.
%D A001563 I. Kortchemski, Asymptotic behavior of permutation records, arXiv: 0804.0446v2 
               [math.CO], 18 May 2008. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Sep 21 2008]
%H A001563 T. D. Noe, <a href="b001563.txt">Table of n, a(n) for n=0..100</a>
%H A001563 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas 
               for Some Functions on Finite Sets</a>
%H A001563 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=30">
               Encyclopedia of Combinatorial Structures 30</a>
%H A001563 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ExponentialIntegral.html">Link to a section of The World of Mathematics.</
               a>
%F A001563 E.g.f.: x/(1-x)^2. a(n)=-A021009(n, 1), n >= 0.
%F A001563 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 (grafix(AT)csl.pl), 
               Oct 22 2007
%F A001563 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 
               (penson(AT)lptl.jussieu.fr), Sep 27 2001
%F A001563 a(0)=0, a(n)=n*a(n-1)+n! - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Feb 16 2003
%F A001563 a(0) = 0, a(n) = (n - 1) * (1 + Sum i=1..n-1 a(i)) for i > 0 - Gerald 
               McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 11 2004
%F A001563 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
%F A001563 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.
%F A001563 a(n) = 1/sum(k!/(n+k+1)!,k=0..infinity), n>0. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Sep 13 2006
%F A001563 a(n)=Sum(k*A143946(n,k),k=1..n(n+1)/2). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Sep 21 2008]
%F A001563 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]
%F A001563 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 R. Janjic (agnus(AT)blic.net), Mar 01 2009]
%F A001563 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%F A001563 a(0) = 0, a(1) = 1 and a(n) = (1+(n-1))^2*product((1+k), k=1..n-2), n=>
               2.
%F A001563 (End)
%e A001563 E_1(x)+gamma+log(x)=x/1-x^2/4+x^3/18-x^4/96+..., x>0.
%p A001563 A001563 := n->n*n!;
%p A001563 seq(sum(mul(j,j=1..n), k=1..n), n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 01 2007
%p A001563 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
%p A001563 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
%p A001563 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
%p A001563 a:=n->add((n!),j=1..n):seq(a(n), n=0..21); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Aug 27 2008]
%p A001563 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]
%p A001563 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]
%t A001563 a = {}; Do[k = CoefficientList[Expand[(y + n!)^n], y]; AppendTo[a, k[[Length[k] 
               - 1]]], {n, 1, 50}]; a - Artur Jasinski (grafix(AT)csl.pl), Oct 22 
               2007
%t A001563 ..and/or..a=s=1;lst={};Do[a=s*n-s;s+=a;AppendTo[lst,a],{n,2*4!}];lst 
               [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 27 2009]
%t A001563 Table[Sum[n!, {i, 1, n}], {n, 0, 21}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 12 2009]
%t A001563 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]
%o A001563 (PARI) a(n)=if(n<0,0,n*n!)
%Y A001563 Cf. A047920, A047922, A000142, A055089, A053495.
%Y A001563 Cf. A123513.
%Y A001563 A143946 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 21 2008]
%Y A001563 A002024 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2008]
%Y A001563 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%Y A001563 Cf. A163931 (E(x,m,n)), A002775 (n^2*n!), A091363 (n^3*n!), A091364 (n^4*n!).
%Y A001563 (End)
%Y A001563 Sequence in context: A081103 A005777 A152392 this_sequence A094304 A094258 
               A086681
%Y A001563 Adjacent sequences: A001560 A001561 A001562 this_sequence A001564 A001565 
               A001566
%K A001563 nonn,easy,nice
%O A001563 0,3
%A A001563 N. J. A. Sloane (njas(AT)research.att.com).