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

Author

Gus Wiseman, Mar 26 2018

Keywords

• nonn

Comments

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

Examples

			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),
()()()()()().
		

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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