cp's OEIS Frontend

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

%I A260504 #16 Mar 03 2025 06:47:17
%S A260504 0,1,2,7,20,91,362,2227,11720,92491,608222,5866147,46290620,527635291,
%T A260504 4857587282,63886537267,672183848720,10019232896491,118594819341542,
%U A260504 1975680877259587,25983971598078020,478434297205284091,6921555837554655002,139581878985127217107
%N A260504 Number of chains in the poset of all odd-sized subsets of {1,2,...,n} ordered by inclusion.
%F A260504 E.g.f.: (s^2 + s*c + s)/(1 - c) where s = sinh(x) and c = cosh(x) - 1.
%F A260504 a(n) ~ n! * (sqrt(3)+2 + (-1)^n*(sqrt(3)-2)) / log(2+sqrt(3))^(n+1). - _Vaclav Kotesovec_, Jul 27 2015
%e A260504 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}}.
%p A260504 # Assuming a(0) = 1:
%p A260504 p := proc(n, z) option remember; local k; if n = 0 then 1 else
%p A260504 normal(add(`if`(k mod 2 = 1, 0, binomial(n, k)*p(k, 0)*(z+1)^(n-k-1)), k=0..n-1))
%p A260504 fi end: A260504 := n -> p(n, 1): seq(A260504(n), n = 0..23); # _Peter Luschny_, Jun 19 2023
%t A260504 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]
%Y A260504 Cf. A260464, A007047.
%K A260504 nonn
%O A260504 0,3
%A A260504 _Geoffrey Critzer_, Jul 27 2015