%I #17 Jul 14 2023 14:30:32

%S 0,0,0,0,0,0,0,1,1,2,2,4,5,8,9,12,14,20,22,29,32,42,47,59,64,81,89,

%T 109,118,144,156,187,202,239,259,303,324,379,408,469,501,577,618,704,

%U 749,851,910,1027,1088,1228,1308,1461,1548,1730,1838,2039,2153,2387

%N Number of partitions of n into 7 squarefree parts.

%H Robert Israel, <a href="/A308952/b308952.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2, where mu is the Möbius function (A008683).

%F a(n) = A308953(n)/n for n > 0.

%p g:= proc(n,k,m) option remember; local i , j;

%p if m=1 then if n=k then return 1 else return 0 fi fi;

%p if k*m < n then return 0 fi;

%p if k*m = n then return 1 fi;

%p add(add(procname(n-i*k,j,m-i), j= select(numtheory:-issqrfree,[$max(1,ceil((n-i*k)/(m-i))) .. k-1])), i=1..min(n/k,m-1));

%p end proc:

%p f:= proc(n) local k;

%p add(g(n,k,7),k=select(numtheory:-issqrfree,[$ceil(n/7)..n]))

%p end proc:

%p f(0):= 0:

%p map(f, [$0..100]); # _Robert Israel_, Jul 03 2019

%t Table[Sum[Sum[Sum[Sum[Sum[Sum[MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

%Y Cf. A008683, A308953, A308954, A308955, A308956, A308957, A308958, A308959, A308960.

%K nonn

%O 0,10

%A _Wesley Ivan Hurt_, Jul 03 2019