|
%I #10 Jul 14 2019 21:09:34
%S 0,0,0,0,0,0,0,0,0,0,10,11,24,26,56,75,128,153,234,304,460,546,770,
%T 943,1296,1550,2054,2430,3220,3770,4830,5642,7168,8283,10302,11935,
%U 14688,16872,20482,23439,28360,32226,38430,43602,51876,58455,68816,77503,90816
%N Sum of all the parts in 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) = 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, where mu is the Möbius function (A008683).
%F a(n) = n * A326626(n).
%F a(n) = A326628(n) + A326629(n) + A326630(n) + A326631(n) + A326632(n) + A326633(n) + A326634(n) + A326635(n) + A326636(n) + A326637(n).
%t Table[n * Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[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, A326628, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326636, A326637.
%K nonn
%O 0,11
%A _Wesley Ivan Hurt_, Jul 14 2019
|
