A326451 Sum of the second largest parts of the partitions of n into 8 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 18, 22, 36, 43, 68, 83, 119, 140, 196, 235, 312, 361, 471, 550, 704, 802, 1008, 1157, 1446, 1643, 2016, 2294, 2798, 3154, 3807, 4285, 5135, 5728, 6797, 7571, 8926, 9880, 11543, 12744, 14827, 16295, 18801, 20645

Offset

0,11

Links

Table of n, a(n) for n=0..51.

Index entries for sequences related to partitions

Formula

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

a(n) = A326444(n) - A326445(n) - A326446(n) - A326447(n) - A326448(n) - A326449(n) - A326450(n) - A326452(n).

Mathematica

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

Crossrefs

Cf. A008683, A326443, A326444, A326445, A326446, A326447, A326448, A326449, A326450, A326452.

Sequence in context: A147617 A308910 A308959 * A263982 A326531 A326636

Adjacent sequences: A326448 A326449 A326450 * A326452 A326453 A326454

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 06 2019

Status

approved