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A007526 a(n) = n(a(n-1) + 1).
(Formerly M3505)
23
0, 1, 4, 15, 64, 325, 1956, 13699, 109600, 986409, 9864100, 108505111, 1302061344, 16926797485, 236975164804, 3554627472075, 56874039553216, 966858672404689, 17403456103284420, 330665665962403999 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Eighteenth- and nineteenth_century combinatorialists call this the number of (nonnull) "variations" of n distinct objects, namely the number of permutations of nonempty subsets of {1,...,n}. Some early references to this sequence are Izquierdo (1659), Caramuel de Lobkowitz (1670), Prestet (1675) and Bernoulli (1713). - Don Knuth, Oct 16 2001, Aug 16 2004

Stirling transform of A006252(n-1)=[0,1,1,2,4,14,38,...] is a(n-1)=[0,1,4,15,64,...]. - Michael Somos, Mar 04 2004

In particular, for n>=1 a(n) is the number of non-empty sequences with n or fewer terms, each a distinct element of {1,...,n}. - Rick L. Shepherd, Jun 08 2005

a(n) = VarScheme(1,n). See A128195 for the definition of VarScheme(k,n). - Peter Luschny, Feb 26 2007

if s(n) is a sequence of the form s(0)=x, s(n)= n(s(n-1)+k), then s(n)= n!*x+a(n)*k. - Gary Detlefs, Jun 06 2010

For n > 0: a(n) = n*A000522(n-1). - Reinhard Zumkeller, Aug 27 2013

REFERENCES

J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas. Freeman, San Francisco, 1974, p. 70.

Jacob Bernoulli, Ars Conjectandi (1713), page 127.

Johannes Caramuel de Lobkowitz, Mathesis Biceps Vetus et Nova (Campania: 1670), volume 2, 942-943.

J. K. Horn, personal communication to Robert G. Wilson v.

Sebastian Izquierdo, Pharus Scientiarum (Lyon: 1659), 327-328.

Jean Prestet, Elemens des Mathematiques (1675), page 341.

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

J. Bernoulli, Wahrscheinlichkeitsrechnung (Ars conjectandi) von Jakob Bernoulli (1713) Uebers. und hrsg. von R. Haussner, Leipzig, W. Engelmann, (1899), [124] Kapitel VII. Variationen ohne Wiederholung. (Page 121).

Peter J. Freyd, Core algebra revisited, Theoretical Computer Science, 375 (2007), Issues 1-3, 193-200.

Z. Kasa and Z. Katai, Scattered subwords and composition of natural numbers, Acta Univ. Sapientiae, Informatica, 4, 2 (2012) 225-236. - From N. J. A. Sloane, Feb 21 2013

Elmar Teufl and Stephan Wagner, Enumeration problems for classes of self-similar graphs, Journal of Combinatorial Theory, Series A, Volume 114, Issue 7, October 2007, Pages 1254-1277.

FORMULA

a(n) = floor( e*n! - 1 ) (J. K. Horn).

a(n) = Sum{r=1..n} nPr = n!*Sum(1/k!, k=0..n-1) = n(a(n-1) + 1).

E.g.f.: x*exp(x)/(1-x). - Vladeta Jovovic, Aug 25 2002

a(n) = sum(k=1, n, k!*C(n, k)). - Benoit Cloitre, Dec 06 2002

a(n) = Sum[n! / k! {k=0...n-1}]. - Ross La Haye, Sep 22 2004

a(n) = sum(k=1..n, prod(j=0..k-1, (n-j) ) ). - Joerg Arndt, Apr 24 2011

Binomial transform of n!-!n. - Paul Barry, May 12 2004

Inverse binomial transform of A066534. - Ross La Haye, Sep 16 2004

Consider the nonempty subsets of the set {1,2,3,...,n} formed by the first n integers. E.g., for n = 3 we have {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. Let the variable sbst denote a subset. For each subset sbst we determine its number of parts, that is nprts(sbst). The sum over all subsets is written as sum_{sbst=subsets}. Then we have A0007526 = sum_{sbst=subsets} nprts(sbst)!. E.g. for n = 3 we have 1!+1!+1!+2!+2!+2!+3! = 15. - Thomas Wieder, Jun 17 2006

For n>0, a(n) = exp(1) * Integral_{x=0..infinity} exp(-exp(x/n)+x) dx. - Gerald McGarvey, Oct 19 2006

a(n) = int(((1+x)^n-1)*exp(-x),x,0,infinity). - Paul Barry, Feb 06 2008

a(n) = GAMMA(n+2)*(1+(-GAMMA(n+1)+exp(1)*GAMMA(n+1, 1))/GAMMA(n+1)). - Thomas Wieder, May 02 2009

E.g.f.: -1/G(0) where G(k)= 1 - 1/(x - x^3/(x^2+(k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jun 10 2012

Conjecture: a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Dec 04 2012

G.f.: (Q(0) - 1)/(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/((1-x)*G(0)) - 1/(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

a(n) = (...((((((0)+1)*1+1)*2+1)*3+1)*4+1)...*n). - Bob Selcoe, Jul 04 2013

G.f.: Q(0)/(2-2*x) - 1/(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 09 2013

G.f.: (W(0) - 1)/(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) = (...(((((0)*1+1)*2+2)*3+3)*4+4)...*n+n). - Bob Selcoe, Apr 30 2014

EXAMPLE

a(3)=15: Let the objects be a, b, and c. The fifteen nonempty ordered subsets are {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}, {abc}, {acb}, {bac}, {bca}, {cab} and {cba}.

MAPLE

A007526 := n->add(n!/k!, k=0..n)-1;

MATHEMATICA

Table[ Sum[n!/(n - r)!, {r, 1, n}], {n, 0, 20}] (* or *) Table[n!*Sum[1/k!, {k, 0, n - 1}], {n, 0, 20}]

a=1; Table[a=(a-1)*(n-1); Abs[a], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Nov 20 2009 *)

FoldList[#1*#2 + #2 &, 0, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)

PROG

(PARI) a(n)=if(n<1, 0, n*(a(n-1)+1))

(PARI) a(n)=if(n<0, 0, n!*polcoeff(x*exp(x+x*O(x^n))/(1-x), n))

(PARI) a(n)= sum(k=1, n, prod(j=0, k-1, (n-j) ) )

(Haskell)

a007526 n = a007526_list !! n

a007526_list = 0 : zipWith (*) [1..] (map (+ 1) a007526_list)

-- Reinhard Zumkeller, Aug 27 2013

CROSSREFS

A000522(n) = a(n)+1.

Row sums of A068424.

Partial sums of A001339.

Cf. A000522, A007526, A001339, A128195.

Sequence in context: A027216 A124541 A134597 * A233536 A097422 A102129

Adjacent sequences:  A007523 A007524 A007525 * A007527 A007528 A007529

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified November 23 00:51 EST 2014. Contains 249836 sequences.