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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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43
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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
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OFFSET
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0,3
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COMMENTS
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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=1..n} k*A092582(n,k). - 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
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. - 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
a(n) is the number of quasilinear weak orderings on {1,...,n}. - J. Devillet, Dec 22 2017
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REFERENCES
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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
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FORMULA
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a(n) = n! * Sum_{k=1..n} 1/k!.
E.g.f.: (exp(x)-1)/(1-x). - Mario Catalani (mario.catalani(AT)unito.it), Feb 06 2003
a(m) = Integral_{s=0..oo} ((1+s)^m - s^m)*exp(-s) = GAMMA(m+1,1) * exp(1) - GAMMA(m+1). - Stephen Crowley, Jul 24 2009
E.g.f.: Q(0)/(1-x), where Q(k)= 1 + (x-1)*k!/(1 - x/(x + (x-1)*(k+1)!/Q(k+1))); (continued fraction). (End)
E.g.f.: x/(1-x)*E(0)/2, where E(k)= 1 + 1/(1 - x/(x + (k+2)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
1/(e - 1) = 1 - 1!/(1*3) - 2!/(3*10) - 3!/(10*41) - 4!/(41*206) - ... (see A056542 and A185108). - Peter Bala, Oct 09 2013
Conjecture: a(n) + (-n-1)*a(n-1) + (n-1)*a(n-2) = 0. - R. J. Mathar, Feb 16 2014
The e.g.f. f(x) = (exp(x)-1)/(1-x) satisfies the differential equation: (1-x)*f'(x) - (2-x)*f(x) + 1, from which we can obtain the recurrence:
a(n+1) = a(n) + n! + Sum_{k=1..n} (n!/k!)*a(k). The above conjectured recurrence can be obtained from the original recurrence or from the differential equation satisfied by f(x). - Emanuele Munarini, Jun 20 2014
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EXAMPLE
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[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, ...]. - Stephen Crowley, Jul 24 2009
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)
x + 3*x^2 + 10*x^3 + 41*x^4 + 206*x^5 + 1237*x^6 + 8660*x^7 + 69281*x^8 + ...
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MAPLE
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add( (n-j)!*binomial(n, j), j=1..n) ;
end proc:
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MATHEMATICA
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RecurrenceTable[{a[0]==0, a[n]==n*a[n-1]+1}, a, {n, 30}] (* Harvey P. Dale, Mar 29 2015 *)
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PROG
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(PARI) a(n)= n!*sum(k=1, n, 1/k!); \\ Joerg Arndt, Apr 24 2011
(Haskell)
a002627 n = a002627_list !! n
a002627_list = 0 : map (+ 1) (zipWith (*) [1..] a002627_list)
(Maxima) makelist(sum(n!/k!, k, 1, n), n, 0, 40); /* Emanuele Munarini, Jun 20 2014 */
(Magma) I:=[1]; [0] cat [n le 1 select I[n] else n*Self(n-1)+1:n in [1..21]]; // Marius A. Burtea, Aug 07 2019
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CROSSREFS
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Conjectured to give records in A130147.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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