A301751 Number of ways to choose a rooted partition of each part in a strict rooted partition of n.

1, 1, 1, 3, 5, 10, 17, 32, 54, 100, 166, 289, 494, 840, 1393, 2400, 3931, 6498, 10861, 17728, 28863, 47557, 77042, 123881, 201172, 322459, 517032, 827993, 1316064, 2084632, 3328204, 5236828, 8247676, 13005652, 20417628, 31934709, 49970815, 77789059, 121144373

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) = 17 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(4)(), (31)(), (22)(), (211)(), (1111)(), (3)(1), (21)(1), (111)(1),
(2)(1)(), (11)(1)().
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A002865, A032305, A063834, A093637, A271619, A281113, A296118, A300383, A301422, A301462, A301467, A301480, A301706.

Programs

Mathematica

nn=50;
ser=x*Product[1+PartitionsP[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

PARI

seq(n)={Vec(prod(k=1, n-1, 1 + numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

Formula

O.g.f.: x * Product_{n > 0} (1 + A000041(n-1) x^n).