OFFSET
0,4
COMMENTS
A set x is full if every element of x is also a subset of x.
Equals the subpartitions of Eulerian numbers (A000295(n)=2^n-n-1); see A115728 for the definition of subpartitions of a partition. - Paul D. Hanna, Jul 03 2006
Also number of transitive rooted identity trees with n branches. - Gus Wiseman, Dec 21 2016
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 123, Problem 20.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..50
Bojan Bašić, Paul Ellis, Dana C. Ernst, Danijela Popović, and Nándor Sieben, Categories of impartial rulegraphs and gamegraphs, arXiv:2312.00650 [math.CO], 2023. See p. 20.
Alberto Casagrande, Carla Piazza, and Alberto Policriti, Is hyper-extensionality preservable under deletions of graph elements?, Elec. Notes Theor. Comp. Sci. (2016) Vol. 322, 103-118.
Richard Peddicord, The number of full sets with n elements, Proc. Amer. Math. Soc., 13 (1962), 825-828.
John Riordan, Letter to N. J. A. Sloane, Jul. 1970
Alexandru Ioan Tomescu, Sets as Graphs, Ph. D. Thesis, Università degli Studi di Udine, Dipartimento di Matematica e Informatica, Dottorato di Ricerca in Informatica, Dec. 2011.
Stephan Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).
Gus Wiseman, Transitive rooted identity trees with n=5 branches
FORMULA
1 = Sum_{n>=0} a(n)*x^n/(1+x)^(2^n). E.g., 1 = 1/(1+x) + 1*x/(1+x)^2 + 1*x^2/(1+x)^4 + 2*x^3/(1+x)^8 + 9*x^4/(1+x)^16 + 88*x^5/(1+x)^32 + 1802*x^6/(1+x)^64 + ... . - Vladeta Jovovic, May 26 2005
Equivalently, a(n) = (-1)^n*C(2^n+n-1, n) - Sum_{k=0..n-1} a(k)*(-1)^(n-k)*C(2^n+2^k+n-k-1, n-k). - Paul D. Hanna, May 26 2005
G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^(2^n-n-1) = 1*(1-x)^0 + 1*x*(1-x)^0 + 1*x^2*(1-x)^1 + 2*x^3*(1-x)^4 + 9*x^3*(1-x)^11 + ... + a(n)*x^n*(1-x)^(2^n-n-1) + ... . - Paul D. Hanna, Jul 03 2006
EXAMPLE
Examples of full sets are 0 := {}, 1 := {0}, 2 := {1,0}, 3a := {2,1,0}, 3b := { {1}, 1, 0}, 4a := { 3a, 2, 1, 0 }.
MAPLE
A001192 := proc(n) option remember: if(n=0)then return 1: fi: return add((-1)^(n-k-1)*binomial(2^k-k, n-k)*procname(k), k=0..n-1); end: seq(A001192(n), n=0..16); # Nathaniel Johnston, Apr 18 2012
MATHEMATICA
max = 16; f[x_] := Sum[a[n]*(x^n/(1+x)^2^n), {n, 0, max}] - 1; cc = CoefficientList[ Series[f[x], {x, 0, max}], x]; Table[a[n], {n, 0, max}] /. First[ Solve[ Thread[cc == 0]]] (* Jean-François Alcover, Nov 02 2011, after Vladeta Jovovic *)
PROG
(PARI) {a(n)=polcoeff(x^n-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(2^k-k-1) ), n)} \\ Paul D. Hanna, Jul 03 2006
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Ryan Propper, Jun 13 2005
STATUS
approved