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

0,10

Links

Robert Israel, Table of n, a(n) for n = 0..2000

Index entries for sequences related to partitions

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}]

Crossrefs

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

Sequence in context: A166515 A339560 A360142 * A302401 A326443 A326522

Adjacent sequences: A308949 A308950 A308951 * A308953 A308954 A308955

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 03 2019

Status

approved