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A052186
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Number of permutations of [n] with no strong fixed points.
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11
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1, 0, 1, 3, 14, 77, 497, 3676, 30677, 285335, 2928846, 32903721, 401739797, 5298600772, 75092880273, 1138261010851, 18378421938366, 314928827507717, 5708689036074089, 109145365739197964, 2195167574579322013, 46331767712354136479, 1023970009016490622478
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OFFSET
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0,4
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COMMENTS
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A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i)<k for i<k and p(j)>k for j>k.
Equals INVERTi transform of the factorials, n starting with 0. Triangle A144108 has row sums = n! with left border = A052186. - Gary W. Adamson, Sep 11 2008
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REFERENCES
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J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178; http://www.combinatorics.org/Volume_18/PDF/v18i1p178.pdf.
Stanley, R., Enumerative Combinatorics, Volume 1 (1986), p. 49
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..200
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]
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FORMULA
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G.f.: F(x)/(1+x*F(x)), F(x) = Sum_{n >= 0} n!*x^n.
a(0)=1, a(1)=0, a(n) = (n-2)*a(n-1) + Sum_{k=0..n-1} a(k)*a(n-1-k) + Sum_{k=0..k-2} a(k)*a(n-2-k) if n>1. - Michael Somos, Oct 11 2006
G.f.: 1/(1-x^2/(1-3x-4x^2/(1-5x-9x^2/(1-7x-16x^2/(1-9x-25x^2/(1-... (continued fraction). - Paul Barry, Dec 09 2009
If p[i]=stirling1(i,1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1) det A. - Milan Janjic, May 08 2010
From Gary W. Adamson, Jul 22 2011: (start) a(n) = upper left term in (-1)*M^(n+1), M = an infinite square production matrix in which a column of (-1)'s is prepended to Pascal's triangle as follows:
-1, 1, 0, 0, 0, 0,...
-1, 1, 1, 0, 0, 0,...
-1, 1, 2, 1, 0, 0,...
-1, 1, 3, 3, 1, 0,...
-1, 1, 4, 6, 4, 1,...
...(end)
G.f.: A(x) =1/(1/G(0) + x) ; G(k)= 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 29 2011
G.f.: A(x)=1/x =1/(1+x)*(1+x/((1+x)*G(0)-x)) ; G(k)= 1 + x*(k+1) - x*(k+2)/G(k+1) ; (continued fraction Euler's kind,1-step ). - Sergei N. Gladkovskii, Dec 29 2011
G.f.: 1/(G(0)+x) where G(k) = 1 - x*(k+1)/(1 - x*(k+1)/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
G.f.: 1/(1-W(0)) where W(k) = x*(2*k+1)-1 - x^2*(k+1)^2/W(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
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MAPLE
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t1 := add(n!*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 0 to 20 do printf(`%d, `, coeff(F, x, i)) od: # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2009
# second Maple program:
a:= proc(n) a(n):= `if`(n<0, -1, -add(a(n-i-1)*i!, i=0..n)) end:
seq(a(n), n=0..25); # Alois P. Heinz, May 21 2013
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MATHEMATICA
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m = 20; CoefficientList[ Series[ 1 / (x + 1/Sum[ n!*x^n, {n, 0, m}]), {x, 0, m}], x] (* Jean-François Alcover, Aug 30 2011, after M. Somos *)
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PROG
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(PARI) {a(n)=if(n<0, 0, polcoeff( 1/ (x+1/sum(k=0, n, k!*x^k, x*O(x^n))), n))} /* Michael Somos, Oct 11 2006 */
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CROSSREFS
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Cf. A006932, A201684.
Cf. A144108, A000142. - Gary W. Adamson, Sep 11 2008
Sequence in context: A198649 A198656 A048779 * A074538 A001564 A059276
Adjacent sequences: A052183 A052184 A052185 * A052187 A052188 A052189
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane, Feb 04 2000
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EXTENSIONS
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Better description from James A. Sellers, Mar 13 2000
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STATUS
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approved
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