A301462 Number of enriched r-trees of size n.

1, 2, 3, 8, 23, 77, 254, 921, 3249, 12133, 44937, 172329, 654895, 2565963, 9956885, 39536964, 156047622, 626262315, 2499486155, 10129445626, 40810378668, 166475139700, 676304156461, 2775117950448, 11342074888693, 46785595997544, 192244951610575, 796245213910406

Offset

0,2

Comments

An enriched r-tree of size n > 0 is either a single node of size n, or a finite sequence of enriched r-trees with weakly decreasing sizes summing to n - 1.

These are different from the R-trees of data science and the enriched R-trees of Bousquet-Mélou and Courtiel.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Formula

O.g.f.: 1/(1 - x) + x Product_{i > 0} 1/(1 - a(i) x^i).

Example

The a(3) = 8 enriched r-trees: 3, (2), ((1)), ((())), (11), (1()), (()1), (()()).

Mathematica

ert[n_]:=ert[n]=1+Sum[Times@@ert/@y, {y, IntegerPartitions[n-1]}];
Array[ert, 30]

Prog

(PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

Crossrefs

Cf. A000081, A003238, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A290689, A300443, A301342-A301345, A301364-A301368, A301422, A301467, A301469, A301470.

Sequence in context: A086628 A365118 A032096 * A120763 A120708 A327009

Adjacent sequences: A301459 A301460 A301461 * A301463 A301464 A301465

Keyword

nonn

Author

Gus Wiseman, Mar 21 2018

Status

approved