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A308952
Number of partitions of n into 7 squarefree parts.
11
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, 109, 118, 144, 156, 187, 202, 239, 259, 303, 324, 379, 408, 469, 501, 577, 618, 704, 749, 851, 910, 1027, 1088, 1228, 1308, 1461, 1548, 1730, 1838, 2039, 2153, 2387
OFFSET
0,10
FORMULA
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).
a(n) = A308953(n)/n for n > 0.
MAPLE
g:= proc(n, k, m) option remember; local i , j;
if m=1 then if n=k then return 1 else return 0 fi fi;
if k*m < n then return 0 fi;
if k*m = n then return 1 fi;
add(add(procname(n-i*k, j, m-i), j= select(numtheory:-issqrfree, [$max(1, ceil((n-i*k)/(m-i))) .. k-1])), i=1..min(n/k, m-1));
end proc:
f:= proc(n) local k;
add(g(n, k, 7), k=select(numtheory:-issqrfree, [$ceil(n/7)..n]))
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Jul 03 2019
MATHEMATICA
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}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 03 2019
STATUS
approved