0,4

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 (pauldhanna(AT)juno.com), Jul 03 2006

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 123, Problem 20.

R. Peddicord, The number of full sets with n elements, Proc. Amer. Math. Soc., 13 (1962), 825-828.

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 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 (pauldhanna(AT)juno.com), Jul 03 2006

Examples of full sets are 0 := {}, 1 := {0}, 2 := {1,0}, 3a := {2,1,0}, 3b := { {1}, 1, 0}, 4a := { 3a, 2, 1, 0 }.

(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 (pauldhanna(AT)juno.com), Jul 03 2006

Cf. A115728, A000295.

Sequence in context: A037172 A135747 A132431 this_sequence A006120 A012941 A059477

Adjacent sequences: A001189 A001190 A001191 this_sequence A001193 A001194 A001195

nonn,nice

njas

More terms from Ryan Propper (rpropper(AT)stanford.edu), Jun 13 2005