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%I #61 Nov 17 2023 11:57:16
%S 1,1,1,2,7,34,206,1476,12123,111866,1143554,12816572,156217782,
%T 2057246164,29111150620,440565923336,7101696260883,121489909224618,
%U 2198572792193786,41966290373704332,842706170872913634,17759399688526009020,391929722837419044420
%N Coefficients of power series A(x) such that n-th term of A(x)^n = n! x^(n-1) for n > 0.
%C Also, number of stablized-interval-free permutations on [n] (see Callan link).
%C Coefficients in the series reversal of the asymptotic expansion of exp(-x)*Ei(x) for x -> inf, where Ei(x) is the exponential integral. - _Vladimir Reshetnikov_, Apr 24 2016
%H Vincenzo Librandi, <a href="/A075834/b075834.txt">Table of n, a(n) for n = 0..100</a>
%H F. Ardila, F. Rincón and L. Williams, <a href="http://arxiv.org/abs/1308.2698">Positroids and non-crossing partitions</a>, arXiv preprint arXiv:1308.2698 [math.CO], 2013.
%H Daniel Birmajer, Juan B. Gil and Michael D. Weiner, <a href="https://arxiv.org/abs/1803.07727">A family of Bell transformations</a>, arXiv:1803.07727 [math.CO], 2018.
%H David Callan, <a href="http://arXiv.org/abs/math.CO/0310157">Counting stabilized-interval-free permutations</a>, arXiv:math/0310157 [math.CO], 2003.
%H David Callan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Callan/callan91.html">Counting Stabilized-Interval-Free Permutations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
%H Colin Defant and Nathan Williams, <a href="https://arxiv.org/abs/2106.05471">Coxeter Pop-Tsack Torsing</a>, arXiv:2106.05471 [math.CO], 2021.
%H Jesse Elliott, <a href="https://arxiv.org/abs/1809.06633">Asymptotic expansions of the prime counting function</a>, arXiv:1809.06633 [math.NT], 2018.
%H Hyungju Park, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Park2/park12.html">An Asymptotic Formula for the Number of Stabilized-Interval-Free Permutations</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.3.
%F a(0)=a(1)=1, a(n) = (n-1)*a(n-1) + Sum_{j=2..n-2}(j-1)*a(j)*a(n-j), n >= 2. - _David Callan_
%F G.f.: A(x) = x/series_reversion(x*G(x)); G(x) = A(x*G(x)); A(x) = G(x/A(x)); where G(x) is the g.f. of the factorials (A000142). - _Paul D. Hanna_, Jul 09 2006
%F G.f.: A(x) = 1 + x/(1 - x*A'(x)/A(x)) = 1 + x/(1-x - x^2*d/dx[(A(x) - 1)/x)]).
%F G.f.: A(x) = 1 + x*F(x) where F(x) satisfies F(x) = 1 + x*F(x) + x^2*F(x)*F'(x) and F'(x) = d/dx F(x). - _Paul D. Hanna_, Sep 02 2008
%F a(n) ~ exp(-1) * n! * (1 - 1/n - 5/(2*n^2) - 32/(3*n^3) - 1643/(24*n^4) - 23017/(40*n^5) - 4215719/(720*n^6)). - _Vaclav Kotesovec_, Feb 22 2014
%F A003319(n+1) = coefficient of x^n in A(x)^n. - _Michael Somos_, Feb 23 2014
%e At n=7, the 7th term of A(x)^7 is 7! x^6, as demonstrated by A(x)^7 = 1 + 7 x + 28 x^2 + 91 x^3 + 294 x^4 + 1092 x^5 + 5040 x^6 + 29093 x^7 + 203651 x^8 + ... .
%e A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 206*x^6 + ... = x/series_reversion(x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 120*x^6 + ...).
%e Related expansions:
%e log(A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 + ...;
%e 1 - x/(A(x) - 1) = x + x^2 + 4*x^3 + 21*x^4 + 136*x^5 + 1030*x^6 +...;
%e (d/dx)((A(x) - 1)/x) = 1 + 4*x + 21*x^2 + 136*x^3 + 1030*x^4 + ... .
%t a = ConstantArray[0,20]; a[[1]]=1; a[[2]]=1; a[[3]]=2; Do[a[[n]] = (n-1)*a[[n-1]] + Sum[(j-1)*a[[j]]*a[[n-j]],{j,2,n-2}],{n,4,20}]; Flatten[{1,a}] (* _Vaclav Kotesovec_ after _David Callan_, Feb 22 2014 *)
%t InverseSeries[Series[Exp[-x] ExpIntegralEi[x], {x, Infinity, 20}]][[3]] (* _Vladimir Reshetnikov_, Apr 24 2016 *)
%o (PARI) a(n)=if(n<0,0,if(n<=1,1,(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j));))
%o (PARI) a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,(k-1)!))))[n+1] \\ _Paul D. Hanna_, Jul 09 2006
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/(1-x*deriv(A)/A));polcoeff(A,n)}
%o (PARI) {a(n)=local(F=1+x*O(x^n)); for(i=0,n,F=1+x*F+x^2*F*deriv(F)+x*O(x^n));polcoeff(1+x*F,n)} \\ _Paul D. Hanna_, Sep 02 2008
%Y Cf. A209881, A091063, A084938.
%Y Cf. A003319.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Oct 14 2002, Jul 30 2008
%E More terms from _David Wasserman_, Jan 26 2005
%E Minor edits by _Vaclav Kotesovec_, Aug 01 2015
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