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

Author

Gus Wiseman, Mar 21 2018

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

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

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

Formula

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