|
| |
|
|
A095989
|
|
INVERTi transform applied to the ordered Bell numbers.
|
|
4
|
|
|
|
1, 2, 8, 48, 368, 3376, 35824, 430512, 5773936, 85482032, 1384936688, 24380214960, 463522810736, 9468048895792, 206831329017328, 4812581925690288, 118843801816575088, 3104590192664327216, 85544737118902122224, 2479681575659312797872, 75434373300016828382576
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
A set composition of n is an ordered sequence [S_1, S_2, ..., S_k] where S_i subset of [n] all disjoint and the union of all S_i is [n] (see A000670). A set composition is atomic if S_1 union ... union S_j does not equal [r] for any r<n and j<k. a(n) is the number of atomic set compositions.
A preference function of n is a word of length n where all the numbers 1 through k occur at least once for some k<=n (see A000670). A preference function is atomic if no strict leading subword contains the only occurrences in the word of the letters 1 through j<k. a(n) is the number of atomic preference functions.
|
|
|
LINKS
|
Alois P. Heinz, Table of n, a(n) for n = 1..400
|
|
|
FORMULA
|
G.f.: 1-1/sum( A000670(k)*q^k, k >= 0)
G.f.: x/(1-2x/(1-2x/(1-4x/(1-3x/(1-6x/(1-4x/(1-8x/(1-5x/(1-...(continued fraction). - Philippe Deléham, Nov 22 2011
G.f.: (1-T(0))/x, where T(k) = 1 - x*(k+1)/(1 - 2*x*(k+1)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 29 2013
Let A(x) be the g.f. A095989, B(x) the g.f. A000670, then A(x)= (1-1/B(x))/x. - Sergei N. Gladkovskii, Nov 29 2013
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Oct 09 2019
|
|
|
EXAMPLE
|
Atomic set compositions a(1)=1: [{1}]; a(2)=2: [{12}], [{2},{1}]; a(3)=8: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}].
Atomic preference functions a(1) = 1: 1; a(2)=2: 11, 21; a(3)=8: 111, 212, 221, 211, 121, 312, 231, 321.
|
|
|
MAPLE
|
A000670 := proc(n) option remember; local k; if n <=1 then 1 else add(binomial(n, k)*A000670(n-k), k=1..n); fi; end: add(A000670(k)*x^k, k=0..20): series(1-1/%, x, 21): [seq(coeff(%, x, i), i=1..20)];
|
|
|
MATHEMATICA
|
max = 20; Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; s = 1 - 1/Sum[ Fubini[k, 1] q^k, {k, 0, max}] + O[q]^max; CoefficientList[s/q, q] (* Jean-François Alcover, Mar 31 2016 *)
|
|
|
CROSSREFS
|
Cf. A000670, A074664, A095993.
Sequence in context: A003576 A225042 A326887 * A177388 A211196 A334856
Adjacent sequences: A095986 A095987 A095988 * A095990 A095991 A095992
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Mike Zabrocki, Jul 18 2004
|
|
|
STATUS
|
approved
|
| |
|
|
