A260504 Number of chains in the poset of all odd-sized subsets of {1,2,...,n} ordered by inclusion.

0, 1, 2, 7, 20, 91, 362, 2227, 11720, 92491, 608222, 5866147, 46290620, 527635291, 4857587282, 63886537267, 672183848720, 10019232896491, 118594819341542, 1975680877259587, 25983971598078020, 478434297205284091, 6921555837554655002, 139581878985127217107

Offset

0,3

Links

Table of n, a(n) for n=0..23.

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))
fi end: A260504 := n -> p(n, 1): seq(A260504(n), n = 0..23); # Peter Luschny, Jun 19 2023

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

Cf. A260464, A007047.

Sequence in context: A118397 A171191 A189771 * A009697 A139012 A132605

Adjacent sequences: A260501 A260502 A260503 * A260505 A260506 A260507

Keyword

nonn

Author

Geoffrey Critzer, Jul 27 2015

Status

approved