A301761 Number of ways to choose a rooted partition of each part in a constant rooted partition of n.

1, 1, 2, 3, 5, 6, 13, 12, 26, 31, 57, 43, 150, 78, 224, 293, 484, 232, 1190, 386, 2260, 2087, 2558, 1003, 11154, 4701, 7889, 13597, 30041, 3719, 83248, 5605, 95006, 84486, 63506, 251487, 654394, 17978, 169864, 490741, 2290336, 37339, 4079503, 53175, 3979370

Offset

1,3

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Formula

a(n) = Sum_{d | n-1} A000041((n-1)/d-1)^d for n > 1. - Andrew Howroyd, Aug 26 2018

Example

The a(7) = 13 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(2)(2), (2)(11), (11)(2), (11)(11),
(1)(1)(1),
()()()()()().

Mathematica

Table[Sum[PartitionsP[n/d-1]^d, {d, Divisors[n]}], {n, 50}]

Prog

(PARI) a(n)=if(n==1, 1, sumdiv(n-1, d, numbpart((n-1)/d-1)^d)) \\ Andrew Howroyd, Aug 26 2018

Crossrefs

Cf. A000005, A000041, A002865, A018818, A063834, A093637, A127524, A260685, A271619, A279787, A301422, A301462, A301467, A301480, A301706.

Sequence in context: A098930 A075372 A064725 * A253644 A100330 A331043

Adjacent sequences: A301758 A301759 A301760 * A301762 A301763 A301764

Keyword

nonn

Author

Gus Wiseman, Mar 26 2018

Status

approved