A326634 - OEIS

%I #10 Jul 14 2019 21:10:27

%S 0,0,0,0,0,0,0,0,0,0,1,1,2,2,5,7,11,14,22,28,42,49,69,84,114,134,179,

%T 212,280,328,424,500,639,742,933,1089,1348,1554,1899,2183,2647,3013,

%U 3598,4096,4867,5493,6464,7289,8521,9557,11098,12418,14368,16013,18393

%N Sum of the fourth largest parts of the partitions of n into 10 squarefree parts.

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

%F a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} mu(r)^2 * mu(q)^2 * mu(p)^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-p-q-r)^2 * k, where mu is the Möbius function (A008683).

%F a(n) = A326627(n) - A326628(n) - A326629(n) - A326630(n) - A326631(n) - A326632(n) - A326633(n) - A326635(n) - A326636(n) - A326637(n).

%t Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k * MoebiusMu[r]^2 * MoebiusMu[q]^2 * MoebiusMu[p]^2 * 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 - p - q - r]^2 , {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

%Y Cf. A008683, A326626, A326627, A326628, A326629, A326630, A326631, A326632, A326633, A326635, A326636, A326637.

%K nonn

%O 0,13

%A _Wesley Ivan Hurt_, Jul 14 2019