0,3

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.

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 (rshepherd2(AT)hotmail.com), Jun 08 2005

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

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

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.

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

J. K. Horn, personal communication to Robert G. Wilson v (rgwv(AT)rgwv.com).

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

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

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).

a(n) = [ 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 (vladeta(AT)eunet.rs), Aug 25 2002

a(n) = sum(k=1, n, k!*C(n, k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 06 2002

Binomial transform of n!-!n. - Paul Barry (pbarry(AT)wit.ie), May 12 2004

Inverse binomial transform of A066534 - Ross La Haye (rlahaye(AT)new.rr.com), Sep 16 2004

a(n) = Sum[n! / k! {k=0...n-1}] - Ross La Haye (rlahaye(AT)new.rr.com), Sep 22 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 (thomas.wieder(AT)t-online.de), Jun 17 2006

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

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

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]

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}.

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

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

(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))

A000522(n)=a(n)+1. Row sums of A068424. Partial sums of A001339.

Cf. A000522, A007526, A001339, A128195.

Adjacent sequences: A007523 A007524 A007525 this_sequence A007527 A007528 A007529

Sequence in context: A027216 A124541 A134597 this_sequence A097422 A102129 A164310

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)