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%I #78 Apr 02 2023 12:27:51
%S 1,1,3,1,13,18,1,50,205,180,1,201,1865,4245,2700,1,875,16674,74165,
%T 114345,56700,1,4138,155477,1208830,3394790,3919860,1587600,1,21145,
%U 1542699,19800165,90265560,182184030,167310360,57153600,1,115973,16385857,335976195,2338275240,7342024200,11471572350,8719666200,2571912000
%N Triangle of coefficients from fractional iteration of e^x - 1.
%C The triangle reflects the Jordan-decomposition of the matrix of Stirling numbers of the second kind. A display of the matrix formula can be found at the Helms link which also explains the generation rule for the A()-numbers in a different way. - _Gottfried Helms_ Apr 19 2014
%C From _Gus Wiseman_, Jan 02 2020: (Start)
%C Also the number of balanced reduced multisystems with atoms {1..n} and depth k. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. For example, row n = 4 counts the following multisystems:
%C {1,2,3,4} {{1},{2,3,4}} {{{1}},{{2},{3,4}}}
%C {{1,2},{3,4}} {{{1},{2}},{{3,4}}}
%C {{1,2,3},{4}} {{{1},{2,3}},{{4}}}
%C {{1,2,4},{3}} {{{1,2}},{{3},{4}}}
%C {{1,3},{2,4}} {{{1,2},{3}},{{4}}}
%C {{1,3,4},{2}} {{{1},{2,4}},{{3}}}
%C {{1,4},{2,3}} {{{1,2},{4}},{{3}}}
%C {{1},{2},{3,4}} {{{1}},{{3},{2,4}}}
%C {{1},{2,3},{4}} {{{1},{3}},{{2,4}}}
%C {{1,2},{3},{4}} {{{1,3}},{{2},{4}}}
%C {{1},{2,4},{3}} {{{1,3},{2}},{{4}}}
%C {{1,3},{2},{4}} {{{1},{3,4}},{{2}}}
%C {{1,4},{2},{3}} {{{1,3},{4}},{{2}}}
%C {{{1}},{{4},{2,3}}}
%C {{{1},{4}},{{2,3}}}
%C {{{1,4}},{{2},{3}}}
%C {{{1,4},{2}},{{3}}}
%C {{{1,4},{3}},{{2}}}
%C (End)
%C From _Harry Richman_, Mar 30 2023: (Start)
%C Equivalently, T(n,k) is the number of length-k chains from minimum to maximum in the lattice of set partitions of {1..n} ordered by refinement. For example, row n = 4 counts the following chains, leaving out the minimum {1|2|3|4} and maximum {1234}:
%C (empty) {12|3|4} {12|3|4} < {123|4}
%C {13|2|4} {12|3|4} < {124|3}
%C {14|2|3} {12|3|4} < {12|34}
%C {1|23|4} {13|2|4} < {123|4}
%C {1|24|3} {13|2|4} < {134|2}
%C {1|2|34} {13|2|4} < {13|24}
%C {123|4} {14|2|3} < {124|3}
%C {124|3} {14|2|3} < {134|2}
%C {134|2} {14|2|3} < {14|23}
%C {1|234} {1|23|4} < {123|4}
%C {12|34} {1|23|4} < {1|234}
%C {13|24} {1|23|4} < {14|23}
%C {14|23} {1|24|3} < {124|3}
%C {1|24|3} < {1|234}
%C {1|24|3} < {13|24}
%C {1|2|34} < {134|2}
%C {1|2|34} < {1|234}
%C {1|2|34} < {12|34}
%C (End)
%C Also the number of cells of dimension k in the fine subdivision of the Bergman complex of the complete graph on n vertices. - _Harry Richman_, Mar 30 2023
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.
%H Alois P. Heinz, <a href="/A008826/b008826.txt">Rows n = 2..150, flattened</a> (first 19 rows from Vincenzo Librandi)
%H Gottfried Helms, <a href="http://mathoverflow.net/questions/133593/151281#151281">How this expression leads to the given sequence</a>, MathOverflow.
%H Federico Ardila and Caroline J. Klivans, <a href="https://doi.org/10.1016/j.jctb.2005.06.004">The Bergman complex of a matroid and phylogenetic trees</a>, J. Combin. Theory Ser. B, 96 (2006), 38-49.
%F G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} Stirling2(n, k)*A(k;x)*x, A(1;x) = 1. - _Vladeta Jovovic_, Jan 02 2004
%F Sum_{k=1..n-1} (-1)^k*T(n,k) = (-1)^(n-1)*(n-1)! = A133942(n-1). - _Geoffrey Critzer_, Sep 06 2020
%e Triangle starts:
%e 1;
%e 1, 3;
%e 1, 13, 18;
%e 1, 50, 205, 180;
%e 1, 201, 1865, 4245, 2700;
%e 1, 875, 16674, 74165, 114345, 56700;
%e 1, 4138, 155477, 1208830, 3394790, 3919860, 1587600;
%e ...
%e The f-vector of (the fine subdivision of) the Bergman complex of the complete graph K_3 is (1, 3). The f-vector of the Bergman complex of K_4 is (1, 13, 18). - _Harry Richman_, Mar 30 2023
%p b:= proc(n) option remember; expand(`if`(n=1, 1,
%p add(Stirling2(n, j)*b(j)*x, j=0..n-1)))
%p end:
%p T:= (n, k)-> coeff(b(n), x, k):
%p seq(seq(T(n, k), k=1..n-1), n=2..10); # _Alois P. Heinz_, Mar 31 2023
%t a[n_, x_] := Sum[ StirlingS2[n, k]*a[k, x]*x, {k, 0, n-1}]; a[1, _] = 1; Table[ CoefficientList[ a[n, x], x] // Rest, {n, 2, 10}] // Flatten (* _Jean-François Alcover_, Dec 11 2012, after _Vladeta Jovovic_ *)
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t tots[m_]:=Prepend[Join@@Table[tots[p],{p,Select[sps[m],1<Length[#]<Length[m]&]}],m];
%t Table[Length[Select[tots[Range[n]],Depth[#]==k&]],{n,2,6},{k,2,n}] (* _Gus Wiseman_, Jan 02 2020 *)
%Y Row sums are A005121.
%Y Alternating row sums are signed factorials A133942(n-1).
%Y Column k = 2 is A008827.
%Y Diagonal k = n - 1 is A006472.
%Y Diagonal k = n - 2 is A059355.
%Y Row n equals row 2^n of A330727.
%Y Cf. A000110, A000111, A000258, A002846, A005121, A008277, A306186, A317176, A318813, A320154, A330667, A330679, A330784.
%K nonn,tabl,nice
%O 2,3
%A _N. J. A. Sloane_, Mar 15 1996
%E More terms from _Vladeta Jovovic_, Jan 02 2004
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