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
Keywords
Examples
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}}.
Programs
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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
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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]
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
Comments