A301767 Number of ways to choose a constant rooted partition of each part in a strict rooted partition of n.

1, 1, 1, 3, 4, 7, 9, 15, 21, 32, 45, 59, 89, 117, 162, 225, 309, 394, 538, 707, 929, 1240, 1613, 2055, 2677, 3517, 4439, 5724, 7288, 9222, 11671, 14809, 18480, 23226, 29138, 36501, 45373, 56438, 69920, 86426, 106715, 131171, 161428, 197717, 242301, 295888

Offset

1,4

Comments

A rooted partition of n is an integer partition of n - 1.

Links

Table of n, a(n) for n=1..46.

Formula

O.g.f.: Product_{n>0} (1 + d(n-1) x^n) where d(n) = A000005(n) and d(0) = 1.

Example

The a(7) = 9 rooted twice-partitions:
(5), (11111),
(4)(), (22)(), (1111)(), (3)(1), (111)(1),
(2)(1)(), (11)(1)().

Mathematica

Table[Sum[Product[If[k===1, 1, DivisorSigma[0, k-1]], {k, ptn}], {ptn, Select[IntegerPartitions[n-1], UnsameQ@@#&]}], {n, 50}]

Crossrefs

Cf. A002865, A032305, A063834, A093637, A279786, A296131, A301422, A301462, A301467, A301480, A301706.

Sequence in context: A241335 A158911 A086772 * A169898 A281734 A086336

Adjacent sequences: A301764 A301765 A301766 * A301768 A301769 A301770

Keyword

nonn

Author

Gus Wiseman, Mar 26 2018

Status

approved