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Number of enriched r-trees of size n.
27

%I #12 Feb 12 2022 20:28:40

%S 1,2,3,8,23,77,254,921,3249,12133,44937,172329,654895,2565963,9956885,

%T 39536964,156047622,626262315,2499486155,10129445626,40810378668,

%U 166475139700,676304156461,2775117950448,11342074888693,46785595997544,192244951610575,796245213910406

%N Number of enriched r-trees of size n.

%C 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.

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

%H Andrew Howroyd, <a href="/A301462/b301462.txt">Table of n, a(n) for n = 0..500</a>

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

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

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

%t Array[ert,30]

%o (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

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

%K nonn

%O 0,2

%A _Gus Wiseman_, Mar 21 2018