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A066534 Total number of walks with length > 0 in the Hasse diagram of a Boolean algebra of order n. 10
0, 1, 6, 30, 152, 840, 5232, 37072, 297600, 2680704, 26812160, 294945024, 3539364864, 46011796480, 644165265408, 9662479226880, 154599668154368, 2628194359738368, 47307498477649920, 898842471080329216 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Let P(A) be the power set of an n-element set A. Then a(n) = the total number of ways to add 1 or more elements of A to each element x of P(A) where the elements to add are not elements of x and order of addition is important. - Ross La Haye, Nov 19 2007
LINKS
Eric Weisstein, Walk
Eric Weisstein, Boolean Algebra
Eric Weisstein, Hasse Diagram
FORMULA
a(n) = n!*Sum_{i+j<n, i, j >= 0} 1/(i!*j!). - Benoit Cloitre, Nov 01 2002
E.g.f.: x*exp(2*x)/(1-x). a(n) = n*(a(n-1)+2^(n-1)). - Vladeta Jovovic, Oct 29 2003
a(n) = Sum_{k=0..n-1} (n! / k!) * 2^k = Sum_{k=0..n-1} P(n, n-k) * 2^k = n! * Sum_{k=0..n-1} 2^k / k! = Sum_{k=1..n} P(n, k) * 2^(n-k) = sum of the n-th row of A090802 from column 1 on = A010842(n) - 2^n = n * A010842(n-1) = binomial transform of A007526 - Ross La Haye, Sep 15 2004
E.g.f.: x/U(0) where U(k) = 1 - 2*x/(2 - 4/(2 + (k+1)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 18 2012
Conjecture: a(n) + (-n-4)*a(n-1) + 4*(n)*a(n-2) + 4*(-n+2)*a(n-3) = 0. - R. J. Mathar, Dec 04 2012
a(n) ~ n! * exp(2). - Vaclav Kotesovec, Jun 01 2013
Mathar's conjectural third-order recurrence above is an easy consequence of Jovovic's first-order recurrence a(n) = n*(a(n-1) + 2^(n-1)). - Peter Bala, Sep 23 2013
EXAMPLE
a(2) = 6 because (2! / 0! * 2^0) + (2! / 1! * 2^1) = 6
MATHEMATICA
a[ n_ ] := n!Sum[ 2^k/k!, {k, 0, n-1} ]
Table[n*Gamma[n, 2]*E^2, {n, 0, 19}] (* Ross La Haye, Oct 09 2005 *)
CROSSREFS
Sequence in context: A054117 A033132 A022023 * A126474 A127017 A152223
KEYWORD
easy,nonn,nice,walk
AUTHOR
Peter Bertok (peter(AT)bertok.com), Jan 07 2002
EXTENSIONS
Edited by Dean Hickerson, Jan 12 2002
Entry revised by Ross La Haye, Aug 18 2006
STATUS
approved

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)