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A008826 Triangle of coefficients from fractional iteration of e^x - 1. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

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)

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.

LINKS

Vincenzo Librandi, Rows n = 2..20, flattened

Gottfried Helms, Answer in mathoverflow.

FORMULA

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

EXAMPLE

Triangle starts:

  1;

  1,  3;

  1, 13,  18;

  1, 50, 205, 180; ...

MATHEMATICA

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-Fran├žois 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 *)

CROSSREFS

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

KEYWORD

nonn,tabl,nice

AUTHOR

N. J. A. Sloane, Mar 15 1996

EXTENSIONS

More terms from Vladeta Jovovic, Jan 02 2004

STATUS

approved

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Last modified May 30 08:04 EDT 2020. Contains 334712 sequences. (Running on oeis4.)