%I #101 Jul 13 2021 19:48:01

%S 1,0,1,3,14,77,497,3676,30677,285335,2928846,32903721,401739797,

%T 5298600772,75092880273,1138261010851,18378421938366,314928827507717,

%U 5708689036074089,109145365739197964,2195167574579322013,46331767712354136479,1023970009016490622478

%N Number of permutations of [n] with no strong fixed points.

%C 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.

%C 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

%D Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49

%H Alois P. Heinz, <a href="/A052186/b052186.txt">Table of n, a(n) for n = 0..450</a>

%H J.-L. Baril, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.

%H Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.

%H Natasha Blitvić and Einar Steingrímsson, <a href="https://arxiv.org/abs/2001.00280">Permutations, moments, measures</a>, arXiv:2001.00280 [math.CO], 2020.

%H Sergey Kitaev and Philip B. Zhang, <a href="https://arxiv.org/abs/1811.07679">Distributions of mesh patterns of short lengths</a>, arXiv:1811.07679 [math.CO], 2018.

%H J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.

%H Richard J. Martin, and Michael J. Kearney, <a href="http://dx.doi.org/10.1007/s00493-014-3183-3">Integral representation of certain combinatorial recurrences</a>, Combinatorica: 35:3 (2015), 309-315.

%H V. Strehl, <a href="/A003149/a003149.pdf">The average number of splitters in a random permutation</a> [Unpublished; included here with the author's permission.]

%F G.f.: F(x)/(1 + x*F(x)), F(x) = Sum_{n >= 0} n!*x^n.

%F 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..n-2} a(k)*a(n-2-k) if n > 1. - _Michael Somos_, Oct 11 2006

%F 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

%F 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

%F From _Gary W. Adamson_, Jul 22 2011: (Start)

%F 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:

%F -1, 1, 0, 0, 0, 0, ...

%F -1, 1, 1, 0, 0, 0, ...

%F -1, 1, 2, 1, 0, 0, ...

%F -1, 1, 3, 3, 1, 0, ...

%F -1, 1, 4, 6, 4, 1, ...

%F ... (End)

%F 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

%F 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

%F 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

%F 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

%F G.f.: 1/(G(0) + x), where G(k)= 1 + x*k - x*(k+1)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 03 2013

%F a(n) ~ n! * (1 - 2/n + 1/n^2 - 1/n^3 - 9/n^4 - 59/n^5 - 474/n^6 - 4560/n^7 - 50364/n^8 - 625385/n^9 - 8622658/n^10), for coefficients see A256168. - _Vaclav Kotesovec_, Mar 16 2015

%F a(n) = n! - Sum_{k=0..n-1} (n-k-1)!*a(k). - _Pontus von Brömssen_, Jul 10 2021

%F a(n) + A006932(n) = n!. - _Pontus von Brömssen_, Jul 10 2021

%p 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_, Mar 22 2009

%p # second Maple program:

%p a:= proc(n) a(n):= -`if`(n<0, 1, add(a(n-i-1)*i!, i=0..n)) end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 21 2013

%t 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 _Michael Somos_ *)

%t nmax = 25; Rest[CoefficientList[Assuming[Element[x, Reals], Series[-1/(ExpIntegralEi[1/x]/E^(1/x) + 1), {x, 0, nmax+1}]], x]] (* _Vaclav Kotesovec_, Aug 05 2015 *)

%o (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 */

%Y Cf. A006932, A201684, A256168.

%Y Cf. A144108, A000142. - _Gary W. Adamson_, Sep 11 2008

%Y Column k=0 of A186373.

%K nonn,easy,nice

%O 0,4

%A _N. J. A. Sloane_, Feb 04 2000

%E Better description from _James A. Sellers_, Mar 13 2000