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%I #12 Feb 03 2021 21:54:06
%S 1,1,1,1,1,2,1,1,3,4,1,1,4,10,9,1,1,5,19,39,20,1,1,6,31,107,160,48,1,
%T 1,7,46,229,647,702,115,1,1,8,64,421,1832,4167,3177,286,1,1,9,85,699,
%U 4191,15583,27847,14830,719,1,1,10,109,1079,8325,44322,137791,191747,70678,1842
%N Array read by antidiagonals: T(n,k) is the number of unlabeled oriented edge-rooted k-gonal 2-trees with n oriented polygons, n >= 0, k >= 2.
%C See section 2 of the Labelle reference.
%H Andrew Howroyd, <a href="/A340814/b340814.txt">Table of n, a(n) for n = 0..1325</a>
%H G. Labelle, C. Lamathe and P. Leroux, <a href="http://arXiv.org/abs/math.CO/0312424">Labeled and unlabeled enumeration of k-gonal 2-trees</a>, arXiv:math/0312424 [math.CO], Dec 23 2003.
%F Column k is the Euler transform of column k+1 of A242249.
%F G.f. of column k: A(x) satisfies A(x) = exp(Sum_{i>0} x^i*A(x^i)^(k-1)/i).
%e Array begins:
%e ============================================================
%e n\k | 2 3 4 5 6 7 8
%e ----+-------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 ...
%e 1 | 1 1 1 1 1 1 1 ...
%e 2 | 2 3 4 5 6 7 8 ...
%e 3 | 4 10 19 31 46 64 85 ...
%e 4 | 9 39 107 229 421 699 1079 ...
%e 5 | 20 160 647 1832 4191 8325 14960 ...
%e 6 | 48 702 4167 15583 44322 105284 220193 ...
%e 7 | 115 3177 27847 137791 487662 1385888 3374267 ...
%e 8 | 286 14830 191747 1255202 5527722 18795035 53275581 ...
%e ...
%o (PARI) \\ here B(n,k) gives g.f. of k-th column.
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o B(n, k)={my(p=1+O(x)); for(n=1, n, p=1+x*Ser(EulerT(Vec(p^(k-1))))); p}
%o { Mat(vector(7, k, Col(B(7, k+1)))) }
%Y Columns k=2..6 are A000081(n+1), A005750(n+1), A052751, A052773, A052781.
%Y Cf. A242249, A340811, A340812.
%K nonn,tabl
%O 0,6
%A _Andrew Howroyd_, Feb 02 2021
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