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A008826
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Triangle of coefficients from fractional iteration of e^x - 1.
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7
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1, 1, 3, 1, 13, 18, 1, 50, 205, 180, 1, 201, 1865, 4245, 2700, 1, 875, 16674, 74165, 114345, 56700, 1, 4138, 155477, 1208830, 3394790, 3919860, 1587600, 1, 21145, 1542699, 19800165, 90265560, 182184030, 167310360, 57153600, 1, 115973
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OFFSET
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2,3
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COMMENTS
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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
From Gus Wiseman, Jan 02 2020: (Start)
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. 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 multisystems:
{1,2,3,4} {{1},{2,3,4}} {{{1}},{{2},{3,4}}}
{{1,2},{3,4}} {{{1},{2}},{{3,4}}}
{{1,2,3},{4}} {{{1},{2,3}},{{4}}}
{{1,2,4},{3}} {{{1,2}},{{3},{4}}}
{{1,3},{2,4}} {{{1,2},{3}},{{4}}}
{{1,3,4},{2}} {{{1},{2,4}},{{3}}}
{{1,4},{2,3}} {{{1,2},{4}},{{3}}}
{{1},{2},{3,4}} {{{1}},{{3},{2,4}}}
{{1},{2,3},{4}} {{{1},{3}},{{2,4}}}
{{1,2},{3},{4}} {{{1,3}},{{2},{4}}}
{{1},{2,4},{3}} {{{1,3},{2}},{{4}}}
{{1,3},{2},{4}} {{{1},{3,4}},{{2}}}
{{1,4},{2},{3}} {{{1,3},{4}},{{2}}}
{{{1}},{{4},{2,3}}}
{{{1},{4}},{{2,3}}}
{{{1,4}},{{2},{3}}}
{{{1,4},{2}},{{3}}}
{{{1,4},{3}},{{2}}}
(End)
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.
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LINKS
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Vincenzo Librandi, Rows n = 2..20, flattened
Gottfried Helms, Answer in mathoverflow.
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FORMULA
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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
Sum_{k=1..n-1} (-1)^k*T(n,k) = (-1)^(n-1)*(n-1)!. - Geoffrey Critzer, Sep 06 2020
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EXAMPLE
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Triangle starts:
1;
1, 3;
1, 13, 18;
1, 50, 205, 180;
...
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MATHEMATICA
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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 *)
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
tots[m_]:=Prepend[Join@@Table[tots[p], {p, Select[sps[m], 1<Length[#]<Length[m]&]}], m];
Table[Length[Select[tots[Range[n]], Depth[#]==k&]], {n, 2, 6}, {k, 2, n}] (* Gus Wiseman, Jan 02 2020 *)
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CROSSREFS
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Row sums are A005121.
Column k = 3 is A008827.
Column k = n is A006472.
Column k = n - 1 is A059355.
Row n is row 2^n of A330727.
Cf. A000110, A000111, A000258, A002846, A008277, A306186, A317176, A318813, A320154, A330667, A330679, A330784.
Sequence in context: A184828 A331998 A053286 * A103440 A116483 A262593
Adjacent sequences: A008823 A008824 A008825 * A008827 A008828 A008829
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KEYWORD
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nonn,tabl,nice
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AUTHOR
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N. J. A. Sloane, Mar 15 1996
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EXTENSIONS
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More terms from Vladeta Jovovic, Jan 02 2004
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STATUS
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approved
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