A308952 Number of partitions of n into 7 squarefree parts.

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

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..2000
• Index entries for sequences related to partitions

Crossrefs

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

Programs

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}]

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.