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A147980 Given a set of positive integers A={1,2,...,n-1,n}, n>=2. Take subsets of A of the form {1,...,n} so only subsets containing numbers 1 and n are allowed. Then a(1)=1 and a(n) is the number of subsets where arithmetic mean of the subset is an integer.

%I A147980 #15 Feb 16 2025 08:33:09
%S A147980 1,0,2,0,4,4,8,12,28,44,84,156,288,540,1020,1904,3616,6860,13024,
%T A147980 24836,47448,90772,174072,334348,643112,1238928,2389956,4615916,
%U A147980 8925808,17278680,33482196,64944060,126083448,244989096,476416560,927167752,1805691728,3519062820
%N A147980 Given a set of positive integers A={1,2,...,n-1,n}, n>=2. Take subsets of A of the form {1,...,n} so only subsets containing numbers 1 and n are allowed. Then a(1)=1 and a(n) is the number of subsets where arithmetic mean of the subset is an integer.
%C A147980 For n odd the value of the arithmetic mean for each possible subset equals (n+1)/2. For n even this value is n/2 or (n+2)/2. If looking after RootMeanSquare for the subset we obtain a sequence [1,0,0,0,0,0,2,...]. We see for example for n=7, A={1,2,3,4,5,6,7} and the only 2 subsets with an integer RootMeanSquare are {1,7}, {1,5,7}. Interestingly the value of RootMeanSquare is 5 for both subsets. So the sequence A140480 RMS numbers is a subsequence of it as a set of divisors of n is clearly a subset of n of the form {1,...,n}.
%H A147980 Alois P. Heinz, <a href="/A147980/b147980.txt">Table of n, a(n) for n = 1..100</a>
%H A147980 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArithmeticMean.html">Arithmetic mean</a>
%e A147980 n=5, A={1,2,3,4,5}. Subsets of A starting with 1 and ending with 5 are : {1,5}, {1,2,5}, {1,3,5}, {1,4,5}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {1,2,3,4,5}. Arithmetic mean of the subset is an integer for subsets : {1,5}, {1,3,5}, {1,2,4,5}, {1,2,3,4,5}. Thus a(5) = 4. The value of the arithmetic mean is 3 for all 4 subsets.
%p A147980 b:= proc(i,s,c) option remember; `if` (i=1, `if` (irem (s, c)=0, 1, 0), b(i-1, s, c)+ b(i-1, s+i, c+1)) end: a:= n-> `if` (n=1, 1, b (n-1, n+1, 2)): seq (a(n), n=1..40);  # _Alois P. Heinz_, May 06 2010
%t A147980 b[i_, s_, c_] := b[i, s, c] = If[i==1, If[Mod[s, c]==0, 1, 0], b[i-1, s, c] + b[i-1, s+i, c+1]];
%t A147980 a[n_] := If[n==1, 1, b[n-1, n+1, 2]];
%t A147980 Array[a, 40] (* _Jean-François Alcover_, Nov 20 2020, after _Alois P. Heinz_ *)
%Y A147980 Cf. A140480.
%K A147980 nonn
%O A147980 1,3
%A A147980 _Ctibor O. Zizka_, Nov 18 2008
%E A147980 More terms from _Alois P. Heinz_, May 06 2010