%I A007526 M3505
%S A007526 0,1,4,15,64,325,1956,13699,109600,986409,9864100,108505111,
%T A007526 1302061344,16926797485,236975164804,3554627472075,56874039553216,
%U A007526 966858672404689,17403456103284420,330665665962403999
%N A007526 a(n) = n(a(n-1) + 1).
%C A007526 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). - D. E. Knuth, Oct 16, 2001; 
               Aug 16 2004.
%C A007526 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
%C A007526 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 (rshepherd2(AT)hotmail.com), Jun 08 2005
%C A007526 a(n) = VarScheme(1,n). See A128195 for the definition of VarScheme(k,
               n). - Peter Luschny (peter(AT)luschny.de), Feb 26 2007
%D A007526 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007526 J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas. Freeman, 
               San Francisco, 1974, p. 70.
%D A007526 Jacob Bernoulli, Ars Conjectandi (1713), page 127.
%D A007526 Johannes Caramuel de Lobkowitz, Mathesis Biceps Vetus et Nova (Campania: 
               1670), volume 2, 942-943.
%D A007526 Peter J. Freyd, Core algebra revisited, Theoretical Computer Science, 
               375 (2007), Issues 1-3, 193-200.
%D A007526 J. K. Horn, personal communication to Robert G. Wilson v (rgwv(AT)rgwv.com).
%D A007526 Sebastian Izquierdo, Pharus Scientiarum (Lyon: 1659), 327-328.
%D A007526 Jean Prestet, Elemens des Mathematiques (1675), page 341.
%H A007526 T. D. Noe, <a href="b007526.txt">Table of n, a(n) for n=0..100</a>
%H A007526 J. Bernoulli, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;
               c=umhistmath;idno=ABZ9501.0001.001">Wahrscheinlichkeitsrechnung (Ars 
               conjectandi) von Jakob Bernoulli (1713) Uebers. und hrsg. von R. 
               Haussner</a>, Leipzig, W. Engelmann, (1899), <a href="http://www.hti.umich.edu/
               t/text/gifcvtdir/abz9501.0001.001/00000307.tifs.gif">[124] Kapitel 
               VII. Variationen ohne Wiederholung. (Page 121)</a>.
%F A007526 a(n) = [ e*n! - 1 ] (J. K. Horn).
%F A007526 a(n) = Sum{r=1..n} nPr = n!*Sum(1/k!, k=0..n-1) = n(a(n-1) + 1).
%F A007526 E.g.f.: x*exp(x)/(1-x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 
               25 2002
%F A007526 a(n) = sum(k=1, n, k!*C(n, k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Dec 06 2002
%F A007526 Binomial transform of n!-!n. - Paul Barry (pbarry(AT)wit.ie), May 12 
               2004
%F A007526 Inverse binomial transform of A066534 - Ross La Haye (rlahaye(AT)new.rr.com), 
               Sep 16 2004
%F A007526 a(n) = Sum[n! / k! {k=0...n-1}] - Ross La Haye (rlahaye(AT)new.rr.com), 
               Sep 22 2004
%F A007526 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 (thomas.wieder(AT)t-online.de), 
               Jun 17 2006
%F A007526 For n>0, a(n) = exp(1) * Integral_{x=0..infinity} exp(-exp(x/n)+x) dx 
               - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 19 2006
%F A007526 a(n)=int(((1+x)^n-1)*exp(-x),x,0,infinity); - Paul Barry (pbarry(AT)wit.ie), 
               Feb 06 2008
%F A007526 a(n)=GAMMA(n+2)*(1+(-GAMMA(n+1)+exp(1)*GAMMA(n+1, 1))/GAMMA(n+1)). [From 
               Thomas Wieder (thomas.wieder(AT)t-online.de), May 02 2009]
%e A007526 a(3)=15: Let the objects be {a, b & c}. The fifteen nonempty ordered 
               subsets are {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}, {abc}, 
               {acb}, {bac}, {bca}, {cab} and {cba}.
%p A007526 A007526 := n->add(n!/k!,k=0..n)-1;
%t A007526 Table[ Sum[n!/(n - r)!, {r, 1, n}], {n, 0, 20}] (* or *) Table[n!*Sum[1/
               k!, {k, 0, n - 1}], {n, 0, 20}]
%t A007526 a=1;Table[a=(a-1)*(n-1);Abs[a],{n,0,40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Nov 20 2009]
%o A007526 (PARI) a(n)=if(n<1,0,n*(a(n-1)+1))
%o A007526 (PARI) a(n)=if(n<0,0,n!*polcoeff(x*exp(x+x*O(x^n))/(1-x),n))
%Y A007526 A000522(n)=a(n)+1. Row sums of A068424. Partial sums of A001339.
%Y A007526 Cf. A000522, A007526, A001339, A128195.
%Y A007526 Sequence in context: A027216 A124541 A134597 this_sequence A097422 A102129 
               A164310
%Y A007526 Adjacent sequences: A007523 A007524 A007525 this_sequence A007527 A007528 
               A007529
%K A007526 nonn,easy
%O A007526 0,3
%A A007526 N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)