A095941 - OEIS

%I #24 May 16 2023 07:03:37

%S 0,0,1,4,13,35,85,194,425,904,1885,3878,7904,16008,32282,64913,130280,

%T 261145,523036,1047017,2095222,4191927,8385695,16773663,33550117,

%U 67103645,134211440,268427907,536861880,1073731053,2147470842,4294952115,8589916646,17179848025

%N Number of subsets of {1,2,...,n} such that every number in the set is no larger than the sum of the other numbers in the set.

%C These are the lengths of the sides of a (possibly degenerate) polygon.

%C Might be called "coalition sets": no member of the set can outnumber all of the others, so a coalition is needed in order to get a majority. - _Jaap Spies_, Jul 14 2004

%H Alois P. Heinz, <a href="/A095941/b095941.txt">Table of n, a(n) for n = 1..3321</a> (first 500 terms from T. D. Noe)

%p b:= proc(n, s) option remember; `if`(s<1, 2^n,

%p      `if`(n*(n+1)/2<s, 0, b(n-1, s)+b(n-1, max(0, s-n))))

%p     end:

%p a:= n-> add(b(j-1, j), j=1..n):

%p seq(a(n), n=1..37);  # _Alois P. Heinz_, Feb 13 2021

%t max = 30; -Accumulate[ Accumulate[q = PartitionsQ[ Range[max]]] + 1] + Accumulate[q] + 2^Range[max] - 1 (* _Jean-François Alcover_, Aug 01 2013, after A095944 *)

%Y See A095944 for formula. Cf. A002623.

%K nonn,nice,easy

%O 1,4

%A Michael Rieck and _W. Edwin Clark_, Jul 13 2004

%E Extended by _Alexander D. Healy_ using data from A095944, Nov 18 2005