This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A099947 #58 May 02 2025 03:07:02
%S A099947 1,1,1,1,2,6,21,85,385,1907,10205,58455,355884,2290536,15518391,
%T A099947 110283179,819675482,6355429550,51293023347,430062712439,
%U A099947 3739408304962,33665192703946,313354708842791,3011545611755271,29847401178719637,304713973031878687,3201007359886598431
%N A099947 Number of topologically connected set partitions of {1,...,n}.
%C A099947 A set partition of {1,...,n} is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y. - _Gus Wiseman_, Feb 19 2019
%H A099947 Alois P. Heinz, <a href="/A099947/b099947.txt">Table of n, a(n) for n = 0..222</a>
%H A099947 Janet Simpson Beissinger, <a href="http://dx.doi.org/10.1016/0097-3165(85)90065-2">The enumeration of irreducible combinatorial objects</a>, J. Comb. Theory, Ser. A, 38 (1985), pp. 143-169. (Example 6.2)
%H A099947 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 A099947 Kenneth J. Dykema, <a href="http://arxiv.org/abs/1602.03469">Generating functions for purely crossing partitions</a>, arXiv:1602.03469 [math.CO], 2016. See column NIS in Table 2 p. 8.
%H A099947 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000925">The number of topologically connected components of a set partition</a>.
%H A099947 Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H A099947 Martin Klazar, <a href="https://kam.mff.cuni.cz/~klazar/bell.pdf">Bell numbers, their relatives and algebraic differential equations</a>
%H A099947 Martin Klazar, <a href="http://dx.doi.org/10.1016/S0097-3165(03)00014-1">Bell numbers, their relatives and algebraic differential equations</a>, J. Combin. Theory, A 102 (2003), 63-87.
%F A099947 From _Paul D. Hanna_, Apr 16 2013: (Start)
%F A099947 O.g.f. A(x) satisfies
%F A099947 (1) A(x) = Sum_{n>=0} A000110(n)*x^n/A(x)^n, where A000110 are the Bell numbers.
%F A099947 (2) A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (A(x) - k*x).
%F A099947 (3) A(x) = 1/(1 - x/(A(x) - 1*x/(1 - x/(A(x) - 2*x/(1 - x/(A(x) - 3*x/(1 - x/(A(x) - 4*x/(1 - x/(A(x) - ... )))))))))), a continued fraction. (End)
%F A099947 B(n) = Sum_p Product_{s in p} a(|s|) where p is a non-crossing set partition of {1,...,n} and B = A000110. In words, every set partition of {1,...,n} can be uniquely decomposed as a non-crossing set partition together with a topologically connected set partition of each block. - _Gus Wiseman_, Feb 19 2019
%e A099947 O.g.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 21*x^6 + 85*x^7 +...
%e A099947 From _Paul D. Hanna_, Apr 16 2013: (Start)
%e A099947 The o.g.f. satisfies
%e A099947 (1) A(x) = 1 + x/A(x) + 2*x^2/A(x)^2 + 5*x^3/A(x)^3 + 15*x^4/A(x)^4 + 52*x^5/A(x)^5 + 203*x^6/A(x)^6 + ... + A000110(n)*x^n/A(x)^n + ...
%e A099947 (2) A(x) = 1 + x/(A(x)-x) + x^2/((A(x)-x)*(A(x)-2*x)) + x^3/((A(x)-x)*(A(x)-2*x)*(A(x)-3*x)) + x^4/((A(x)-x)*(A(x)-2*x)*(A(x)-3*x)*(A(x)-4*x)) + ... (End)
%e A099947 From _Gus Wiseman_, Feb 19 2019: (Start)
%e A099947 The a(1) = 1 through a(6) = 21 topologically connected set partitions:
%e A099947 {{1}} {{12}} {{123}} {{1234}} {{12345}} {{123456}}
%e A099947 {{13}{24}} {{124}{35}} {{1235}{46}}
%e A099947 {{13}{245}} {{124}{356}}
%e A099947 {{134}{25}} {{1245}{36}}
%e A099947 {{135}{24}} {{1246}{35}}
%e A099947 {{14}{235}} {{125}{346}}
%e A099947 {{13}{2456}}
%e A099947 {{134}{256}}
%e A099947 {{1345}{26}}
%e A099947 {{1346}{25}}
%e A099947 {{135}{246}}
%e A099947 {{1356}{24}}
%e A099947 {{136}{245}}
%e A099947 {{14}{2356}}
%e A099947 {{145}{236}}
%e A099947 {{146}{235}}
%e A099947 {{15}{2346}}
%e A099947 {{13}{25}{46}}
%e A099947 {{14}{25}{36}}
%e A099947 {{14}{26}{35}}
%e A099947 {{15}{24}{36}}
%e A099947 (End)
%t A099947 a[0] = 1; a[n_] := Module[{A = 1 + x}, For[i = 1, i <= n, i++, A = Sum[x^m/Product[A - k*x + x*O[x]^n, {k, 1, m}], {m, 0, n}]]; Coefficient[A, x^n]]; Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Sep 13 2013, after _Paul D. Hanna_ *)
%t A099947 nn=8;
%t A099947 nonXQ[stn_]:=!MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A099947 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A099947 Solve[Table[BellB[n]==Sum[Product[a[Length[s]],{s,stn}],{stn,Select[sps[Range[n]],nonXQ]}],{n,nn}],Array[a,nn]] (* _Gus Wiseman_, Feb 19 2019 *)
%o A099947 (PARI) {a(n)=if(n<0, 0, polcoeff( x/serreverse(x*serlaplace(exp(exp(x+x*O(x^n))-1))), n))} /* _Michael Somos_, Sep 22 2005 */
%o A099947 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/prod(k=1, m, A - k*x +x*O(x^n)) )); polcoeff(A, n)} \\ _Paul D. Hanna_, Apr 16 2013
%Y A099947 Cf. A000108, A000110, A001187, A007297, A016098, A092918, A268814, A268815, A305078, A306438, A323818, A324166, A324172, A324173.
%K A099947 nonn,easy,nice
%O A099947 0,5
%A A099947 _N. J. A. Sloane_, Nov 12 2004
%E A099947 Name edited by _Gus Wiseman_, Feb 19 2019