A301762 Number of ways to choose a constant rooted partition of each part in a rooted partition of n.

1, 1, 2, 4, 7, 12, 21, 34, 55, 90, 143, 220, 347, 528, 805, 1226, 1831, 2719, 4048, 5940, 8710, 12714, 18403, 26529, 38220, 54679, 77899, 110810, 156848, 221181, 311635, 436705, 610597, 852125, 1184928, 1644136, 2276551, 3142523, 4328960, 5953523, 8167209

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(5) = 7 rooted twice-partitions where the latter rooted partitions are constant: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().
		

Crossrefs

Cf. A002865, A063834, A093637, A279784, A295935, A300383, A301422, A301462, A301467, A301480, A301706.

Programs

Mathematica

Table[Sum[Product[If[k===1,1,DivisorSigma[0,k-1]],{k,ptn}],{ptn,IntegerPartitions[n-1]}],{n,20}]

Formula

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