A301754 Number of ways to choose a strict rooted partition of each part in a strict rooted partition of n.

1, 1, 1, 2, 3, 5, 8, 13, 18, 29, 44, 67, 100, 150, 217, 326, 470, 690, 1011, 1463, 2099, 3049, 4355, 6214, 8886, 12632, 17885, 25377, 35763, 50252, 70942, 99246, 138600, 193912, 270286, 375471, 522224, 723010, 1000435, 1383002, 1907724, 2624492, 3613885

Offset

1,4

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 + A000009(n-1) x^n).

Example

The a(8) = 13 rooted twice-partitions:
(6), (51), (42), (321),
(5)(), (41)(), (32)(), (4)(1), (31)(1), (3)(2), (21)(2),
(3)(1)(), (21)(1)().

Mathematica

nn=50;
ser=x*Product[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(prod(k=1, n-1, 1 + u[k]*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

Crossrefs

Cf. A002865, A032305, A063834, A093637, A196545, A279785, A296120, A301422, A301462, A301467, A301480, A301706.

Sequence in context: A081612 A343484 A374737 * A120761 A355513 A281967

Adjacent sequences: A301751 A301752 A301753 * A301755 A301756 A301757

Keyword

nonn

Author

Gus Wiseman, Mar 26 2018

Status

approved