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A007526
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a(n) = n(a(n-1) + 1).
(Formerly M3505)
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21
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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; internal format)
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OFFSET
| 0,3
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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). - 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
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 [From Gary Detlefs (gdetlefs(AT)aol.com), Jun 06 2010]
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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.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
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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).
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FORMULA
| 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
a(n) = Sum[n! / k! {k=0...n-1}] - Ross La Haye (rlahaye(AT)new.rr.com), 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 (pbarry(AT)wit.ie), May 12 2004
Inverse binomial transform of A066534 - Ross La Haye (rlahaye(AT)new.rr.com), 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 (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]
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EXAMPLE
| 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}.
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MAPLE
| A007526 := n->add(n!/k!, k=0..n)-1;
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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}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 20 2009]
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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) ) )
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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 * A097422 A102129 A164310
Adjacent sequences: A007523 A007524 A007525 * A007527 A007528 A007529
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
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![[124] Kapitel VII. Variationen ohne Wiederholung. (Page 121)](http://www.hti.umich.edu/t/text/gifcvtdir/abz9501.0001.001/00000307.tifs.gif){kind=link}