A301753 Number of ways to choose a strict rooted partition of each part in a rooted partition of n.

1, 1, 2, 3, 6, 9, 16, 25, 43, 66, 108, 166, 269, 408, 643, 975, 1517, 2277, 3497, 5223, 7936, 11803, 17736, 26219, 39174, 57594, 85299, 124957, 183987, 268158, 392685, 569987, 830282, 1200843, 1740422, 2507823, 3620550, 5197885, 7472229, 10694865, 15319700

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..500

Formula

O.g.f.: x * Product_{n > 0} 1/(1 - A000009(n-1) x^n).

Example

The a(7) = 16 rooted twice-partitions:
(5), (32), (41),
(2)(2), (3)(1), (4)(), (21)(1), (31)(),
(1)(1)(1), (2)(1)(), (3)()(), (21)()(),
(1)(1)()(), (2)()()(),
(1)()()()(),
()()()()()().

Mathematica

nn=50;
ser=x*Product[1/(1-PartitionsQ[n-1]x^n), {n, nn}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, nn}]

Prog

(PARI) seq(n)={my(u=Vec(prod(k=1, n-1, 1 + x^k + O(x^n)))); Vec(1/prod(k=1, n-1, 1 - u[k]*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

Crossrefs

Cf. A002865, A032305, A063834, A093637, A270995, A281113, A296119, A301422, A301462, A301467, A301480, A301706, A301751.

Sequence in context: A319642 A062114 A094768 * A275548 A260710 A359994

Adjacent sequences: A301750 A301751 A301752 * A301754 A301755 A301756

Keyword

nonn

Author

Gus Wiseman, Mar 26 2018

Status

approved