OFFSET
0,3
FORMULA
E.g.f.: (s^2 + s*c + s)/(1 - c) where s = sinh(x) and c = cosh(x) - 1.
a(n) ~ n! * (sqrt(3)+2 + (-1)^n*(sqrt(3)-2)) / log(2+sqrt(3))^(n+1). - Vaclav Kotesovec, Jul 27 2015
EXAMPLE
a(4) = 20 because there are C(4,1) + C(4,3) = 8 chains of length zero (these are the odd-sized subsets of {1,2,3,4}. There are 12 chains of length one: {{1},{1,2,3}}; {{1},{1,2,4}}; {{1},{1,3,4}}; {{2},{1,2,3}}; {{2},{1,2,4}}; {{2},{2,3,4}}; {{3},{1,2,3}}; {{3},{1,3,4}}; {{3},{2,3,4}}; {{4},{1,2,4}}; {{4},{1,3,4}}; {{4},{2,3,4}}.
MAPLE
# Assuming a(0) = 1:
p := proc(n, z) option remember; local k; if n = 0 then 1 else
normal(add(`if`(k mod 2 = 1, 0, binomial(n, k)*p(k, 0)*(z+1)^(n-k-1)), k=0..n-1))
MATHEMATICA
nn = 20; c=Cosh[x]-1; s=Sinh[x]; Range[0, nn]!CoefficientList[Series[(s^2 + s c + s)/(1 - c), {x, 0, nn}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jul 27 2015
STATUS
approved