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%I #37 Apr 15 2024 13:16:44
%S 1,1,3,13,69,421,2867,21477,175769,1567273,15213955,160727997,
%T 1846282381,23013527421,310284575683,4506744095141,70199956070705,
%U 1167389338452753,20636801363971139,386304535988493101,7630926750477398037,158584458024427667669
%N G.f. satisfies: A(x) = Sum_{n>=0} n!*x^n*A(x)^n.
%C a(n) = number of partitions of [n] into sets of noncrossing lists. For example, a(4) = 69 counts the 73 partitions of [n] into sets of lists (A000262) except for 13-24, 13-42, 31-24, 31-42 which are crossing. - _David Callan_, Jul 25 2008
%H Alois P. Heinz, <a href="/A088368/b088368.txt">Table of n, a(n) for n = 0..200</a>
%H David Callan, <a href="http://arxiv.org/abs/0711.4841">Sets, Lists and Noncrossing Partitions</a>, arXiv:0711.4841 [math.CO], 2007-2008.
%H David Callan and Emeric Deutsch, <a href="http://arxiv.org/abs/1112.3639">The Run Transform</a>, arXiv preprint arXiv:1112.3639 [math.CO], 2011.
%H Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 11.
%F G.f.: A(x) = (1/x)*Series_Reversion( x/[Sum_{n>=0} n!*x^n] ).
%F G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - 2*x*A(x)/(1 - 3*x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - ...)))))))), a recursive continued fraction.
%F G.f. satisfies: A(x/F(x)) = F(x) where F(x) = Sum_{n>=0} n!*x^n.
%F G.f. A(x) satisfies: A = 1 + x*A(x) * (x*A(x)^2)' / (x*A(x))'. - _Paul D. Hanna_, Apr 01 2018
%F a(n) ~ exp(1) * n!. - _Vaclav Kotesovec_, Apr 10 2019
%e G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 421*x^5 + 2867*x^6 +...
%e where
%e A(x) = 1 + x*A(x) + 2!*x^2*A(x)^2 + 3!*x^3*A(x)^3 + 4!*x^4*A(x)^4 +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 173*x^4 + 1058*x^5 +...
%e A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 321*x^4 + 1977*x^5 +...
%e A(x)^4 = 1 + 4*x + 18*x^2 + 92*x^3 + 523*x^4 + 3256*x^5 +...
%e A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 790*x^4 + 4986*x^5 +...
%t FrequencyDistribution[list_List] := Module[{set = Union[list]}, Table[{set[[i]], Count[list, set[[i]]]}, {i, Length[set]}]]; a[0] = 1; a[n_]/;n>=1 := a[n] = Apply[Plus,Module[{frequencies},Map[(frequencies=Map[Last,FrequencyDistribution[ # ]]; Sum[frequencies]!*Apply[Multinomial,frequencies]* Product[Map[a,# ]])&,Partitions[n]-1 ]]] Table[a[n],{n,0,15}] - _David Callan_, Jul 25 2008
%o (PARI) {a(n)=polcoeff(1/x*serreverse(x/sum(m=0,n,m!*x^m)+x^2*O(x^n)),n)}
%o for(n=0, 30, print1(a(n),", "))
%o (PARI) /* Recursive continued fraction: */
%o {a(n)=local(A=1+x,CF=1+x*O(x^(n+2))); for(i=1,n, for(k=1, n+1, CF=1/(1-((n-k+1)\2+1)*x*A*CF));A=CF); polcoeff(A,n)}
%o for(n=0, 30, print1(a(n),", "))
%o (PARI) /* Differential Equation */
%o {a(n) = my(A=1+x); for(i=0,n, A = 1 + x*A*(x*A^2)'/(x*A+ x^2*O(x^n))' ); polcoeff(A,n)}
%o for(n=0, 30, print1(a(n),", ")) \\ _Paul D. Hanna_, Apr 01 2018
%Y Cf. A198916.
%K nonn,changed
%O 0,3
%A _Paul D. Hanna_, Sep 28 2003
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