A001563 - OEIS

%I M3545 N1436 #228 Jan 03 2025 12:23:33

%S 0,1,4,18,96,600,4320,35280,322560,3265920,36288000,439084800,

%T 5748019200,80951270400,1220496076800,19615115520000,334764638208000,

%U 6046686277632000,115242726703104000,2311256907767808000,48658040163532800000,1072909785605898240000

%N a(n) = n*n! = (n+1)! - n!.

%C 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 Denominators in power series expansion of E_1(x) + gamma + log(x), x > 0. - _Michael Somos_, Dec 11 2002

%C 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., it makes no change. Permutation 1 is (0,1,3,2), which rotates the last 2 elements. Permutation 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 Stirling transform of a(n+1)=[4,18,96,600,...] is A083140(n+1)=[4,22,154,...]. - _Michael Somos_, Mar 04 2004

%C From _Michael Somos_, Apr 27 2012: (Start)

%C Stirling transform of a(n)=[1,4,18,96,...] is A069321(n)=[1,5,31,233,...].

%C Partial sums of a(n)=[0,1,4,18,...] is A033312(n+1)=[0,1,5,23,...].

%C Binomial transform of A000166(n+1)=[0,1,2,9,...] is a(n)=[0,1,4,18,...].

%C Binomial transform of A000255(n+1)=[1,3,11,53,...] is a(n+1)=[1,4,18,96,...].

%C Binomial transform of a(n)=[0,1,4,18,...] is A093964(n)=[0,1,6,33,...].

%C Partial sums of A001564(n)=[1,3,4,14,...] is a(n+1)=[1,4,18,96,...].

%C (End)

%C 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=0..n-1}k*A123513(n,k). - _Emeric Deutsch_, Oct 02 2006

%C Equivalently, in the notation of David, Kendall and Barton, p. 263, this is the total number of consecutive ascending pairs in all permutations on n+1 letters (cf. A010027). - _N. J. A. Sloane_, Apr 12 2014

%C 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

%C Number of factors in a determinant when writing down all multiplication permutations. - _Mats Granvik_, Sep 12 2008

%C 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. - _Emeric Deutsch_, Sep 21 2008

%C Equals eigensequence of triangle A002024 ("n appears n times"). - _Gary W. Adamson_, Dec 29 2008

%C 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). - _Gary W. Adamson_, Apr 17 2009

%C Row lengths of the triangle in A030298. - _Reinhard Zumkeller_, Mar 29 2012

%C a(n) is also the number of minimum (n-)distinguishing labelings of the star graph S_{n+1} on n+1 nodes. - _Eric W. Weisstein_, Oct 14 2014

%C When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the right, i.e., a(n) is the permutation with the cycle notation (0 1 ... n-1 n). Compare array A051683 for circular shifts to the right in a broader sense. Compare sequence A007489 for circular shifts to the left. - _Tilman Piesk_, Apr 29 2017

%C a(n-1) is the number of permutations on n elements with no cycles of length n. - _Dennis P. Walsh_, Oct 02 2017

%C The number of pandigital numbers in base n+1, such that each digit appears exactly once. For example, there are a(9) = 9*9! = 3265920 pandigital numbers in base 10 (A050278). - _Amiram Eldar_, Apr 13 2020

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 218.

%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 336.

%D F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 37, equation 37:6:1 at page 354.

%H T. D. Noe, <a href="/A001563/b001563.txt">Table of n, a(n) for n = 0..100</a>

%H J. D. H. Dickson, <a href="/A002775/a002775.pdf">Discussion of two double series arising from the number of terms in determinants of certain forms</a>, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]

%H Loïc Foissy, <a href="https://arxiv.org/abs/2406.01120">The antipode of of [sic] a Com-PreLie Hopf algebra</a>, arXiv:2406.01120 [math.CO], 2024. See p. 9.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=30">Encyclopedia of Combinatorial Structures 30</a>.

%H Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.

%H L. B. W. Jolley, <a href="https://archive.org/details/summationofserie00joll">Summation of Series</a>, Dover, 1961.

%H I. Kortchemski, <a href="http://arxiv.org/abs/0804.0446">Asymptotic behavior of permutation records</a>, arXiv: 0804.0446v2 [math.CO], 18 May 2008.

%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages).

%H Rezsö L. Lovas and István Mezö, <a href="https://arxiv.org/abs/1008.0713">On an exotic topology of the integers</a>, arXiv:1008.0713 [math.GN], 2010. See p. 4.

%H Daniel J. Mundfrom, <a href="http://dx.doi.org/10.1006/eujc.1994.1057">A problem in permutations: the game of `Mousetrap'</a>, European J. Combin. 15 (1994), no. 6, 555-560.

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

%H J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].

%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages).

%H A. van Heemert, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002177749">Cyclic permutations with sequences and related problems</a>, J. Reine Angew. Math., 198 (1957), 56-72.

%H Dennis P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/NOKCYCLB.pdf">The number of permutations with no k-cycles</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinguishingNumber.html">Distinguishing Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">Exponential Integral</a>.

%F From _Michael Somos_, Dec 11 2002: (Start)

%F E.g.f.: x / (1 - x)^2.

%F a(n) = -A021009(n, 1), n >= 0. (End)

%F 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

%F Integral representation as n-th moment of a function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*(x*(x-1)*exp(-x)) dx, for n>=0. This representation may not be unique. - _Karol A. Penson_, Sep 27 2001

%F a(0)=0, a(n) = n*a(n-1) + n!. - _Benoit Cloitre_, Feb 16 2003

%F a(0) = 0, a(n) = (n - 1) * (1 + Sum_{i=1..n-1} a(i)) for i > 0. - _Gerald McGarvey_, Jun 11 2004

%F Arises in the denominators of the following identities: Sum_{n>=1} 1/(n*(n+1)*(n+2)) = 1/4, Sum_{n>=1} 1/(n*(n+1)*(n+2)*(n+3)) = 1/18, Sum_{n>=1} 1/(n*(n+1)*(n+2)*(n+3)*(n+4)) = 1/96, etc. The general expression is Sum_{n>=k} 1/C(n, k) = k/(k-1). - Dick Boland, Jun 06 2005 [And the general expression implies that Sum_{n>=1} 1/(n*(n+1)*...*(n+k-1)) = (Sum_{n>=k} 1/C(n, k))/k! = 1/((k-1)*(k-1)!) = 1/a(k-1), k >= 2. - _Jianing Song_, May 07 2023]

%F a(n) = Sum_{m=2..n+1} |Stirling1(n+1, m)|, n >= 1 and a(0):=0, where Stirling1(n, m) = A048994(n, m), n >= m = 0.

%F a(n) = 1/(Sum_{k>=0} k!/(n+k+1)!), n > 0. - _Vladeta Jovovic_, Sep 13 2006

%F a(n) = Sum_{k=1..n(n+1)/2} k*A143946(n,k). - _Emeric Deutsch_, Sep 21 2008

%F 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. - Peter Breznay (breznayp(AT)uwgb.edu), Sep 28 2008

%F If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)* Stirling2(j,i)*x^(k-j) then a(n) = (-1)^(n-1)*f(n,1,-2), (n >= 1). - _Milan Janjic_, Mar 01 2009

%F Sum_{n>=1} (-1)^(n+1)/a(n) =  0.796599599... [Jolley eq. 289]

%F 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

%F G.f.: W(0)*(1-sqrt(x)) - 1, where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+2)/(sqrt(x)*(k+2) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 18 2013

%F G.f.: T(0)/x - 1/x, where T(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 - (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 17 2013

%F G.f.: Q(0)*(1-x)/x - 1/x, where Q(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/Q(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Oct 22 2013

%F D-finite with recurrence: a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - _R. J. Mathar_, Jan 14 2020

%F a(n) = (-1)^(n+1)*(n+1)*Sum_{k=1..n} A094485(n,k)*Bernoulli(k). The inverse of the Worpitzky representation of the Bernoulli numbers. - _Peter Luschny_, May 28 2020

%F From _Amiram Eldar_, Aug 04 2020: (Start)

%F Sum_{n>=1} 1/a(n) = Ei(1) - gamma = A229837.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = gamma - Ei(-1) = A239069. (End)

%F a(n) = Gamma(n)*A000290(n) for n > 0. - _Jacob Szlachetka_, Jan 01 2022

%e E_1(x) + gamma + log(x) = x/1 - x^2/4 + x^3/18 - x^4/96 + ..., x > 0. - _Michael Somos_, Dec 11 2002

%e G.f. = x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 4320*x^6 + 35280*x^7 + 322560*x^8 + ...

%p A001563 := n->n*n!;

%t Table[n!n,{n,0,25}] (* _Harvey P. Dale_, Oct 03 2011 *)

%o (PARI) {a(n) = if( n<0, 0, n * n!)} /* _Michael Somos_, Dec 11 2002 */

%o (Haskell)

%o a001563 n = a001563_list !! n

%o a001563_list = zipWith (-) (tail a000142_list) a000142_list

%o -- _Reinhard Zumkeller_, Aug 05 2013

%o (Magma) [Factorial(n+1)-Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Aug 08 2014

%o (Sage) [n*factorial(n) for n in (0..20)] # _G. C. Greubel_, Dec 30 2019

%o (GAP) List([0..20], n-> n*Factorial(n) ); # _G. C. Greubel_, Dec 30 2019

%Y Cf. A000142, A002024, A047920, A047922, A053495, A055089, A123513, A143946, A229837, A239069.

%Y Cf. A163931 (E(x,m,n)), A002775 (n^2*n!), A091363 (n^3*n!), A091364 (n^4*n!).

%Y Cf. sequences with formula (n + k)*n! listed in A282466.

%Y Row sums of A185105, A322383, A322384, A094485.

%K nonn,easy,nice,changed

%O 0,3

%A _N. J. A. Sloane_