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A002627
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a(n) = n*a(n-1) + 1, a(0) = 0.
(Formerly M2858 N1149)
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18
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0, 1, 3, 10, 41, 206, 1237, 8660, 69281, 623530, 6235301, 68588312, 823059745, 10699776686, 149796873605, 2246953104076, 35951249665217, 611171244308690, 11001082397556421, 209020565553572000, 4180411311071440001, 87788637532500240022
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| This sequence shares divisibility properties with A000522; each of the primes in A072456 divide only a finite number of terms of this sequence. - T. D. Noe, Jul 07 2005
Sum of the lengths of the first runs in all permutations of [n]. Example: a(3)=10 because the lengths of the first runs in the permutation (123),(13)2,(3)12,(2)13,(23)1 and (3)21 are 3,2,1,1,2 and 1, respectively (first runs are enclosed between parentheses). Number of cells in the last columns of all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. a(n)=Sum(k*A092582(n,k), k=1..n). - Emeric Deutsch, Aug 16 2006
Starting with offset 1 = eigensequence of an infinite lower triangular matrix with (1, 2, 3,...) as the right border, (1, 1, 1,...) as the left border, and the rest zeros. [Gary W. Adamson, Apr 27 2009]
Sums of rows of the triangle in A173333, n>0. [From Reinhard Zumkeller, Feb 19 2010]
if s(n) is a sequence defined as s(0)=x,s(n)=n*s(n-1)+k,n>0 then s(n)=n!*x + a(n)*k. [From Gary Detlefs, Feb 20 2010]
Number of arrangements of proper subsets of n distinct objects, i.e. arrangements which are not permutations (where the empty set is considered a proper subset of any nonempty set); see example. - Daniel Forgues, Apr 23 2011
For n>=0, A002627(n+1) is the sequence of sums of Pascal-like triangle with one side 1,1,..., and the other side A000522. - Vladimir Shevelev, Feb 06 2012
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REFERENCES
| E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 150
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FORMULA
| a(n) = n! * sum(k=1..n, 1/k!).
a(n) = floor( n! * (e-1) ), n>=1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 08 2002
E.g.f.: (exp(x)-1)/(1-x) - Mario Catalani (mario.catalani(AT)unito.it), Feb 06 2003
Binomial transform of A002467. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004
a(n) = sum(j=1..n, (n-j)!*binomial(n,j)). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 31 2006
a(n) = 1+sum(k=0,n-1, k*a(k)) - Benoit Cloitre, Jul 26 2008
a(m) = int(((1+s)^m-s^m)*exp(-s),s=0..infinity) = GAMMA(m+1,1) * exp(1) -GAMMA(m+1) [From Stephen Crowley, Jul 24 2009]
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EXAMPLE
| [a(0),a(1),..]=GAMMA(m+1,1)*exp(1)-GAMMA(m+1)=[exp(-1)*exp(1)-1, 2*exp(-1)*exp(1)-1, 5*exp(-1)*exp(1)-2, 16*exp(-1)*exp(1)-6, 65*exp(-1)*exp(1)-24, 326*exp(-1)*exp(1)-120,...] [From Stephen Crowley, Jul 24 2009]
From Daniel Forgues, April 25, 2011: (Start)
n=0: {}: #{} = 0
n=1: {1}: #{()} = 1
n=2: {1,2}: #{(),(1),(2)} = 3
n=3: {1,2,3}: #{(),(1),(2),(3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)} = 10
(End)
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MAPLE
| a:= n-> add((n-j)! *binomial(n, j), j=1..n): seq(a(n), n=0..21); # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 31 2006
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MATHEMATICA
| FoldList[ #1*#2 + 1 &, 0, Range[21]] (from Robert G. Wilson v, Oct 11 2005)
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PROG
| (Pari) a(n)= n!*sum(k=1, n, 1/k!); [Joerg Arndt, Apr 24 2011]
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CROSSREFS
| A002627(n) = A000522(n) - n!. Second diagonal of A059922, Cf. A056542.
Cf. A092582.
Sequence in context: A116540 A000248 A030927 * A030802 A030942 A030855
Adjacent sequences: A002624 A002625 A002626 * A002628 A002629 A002630
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Comments from Michael Somos
More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 16 2006
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