%I #11 Feb 03 2021 21:51:47
%S 1,1,1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,3,7,6,1,1,1,3,11,18,11,1,1,1,4,
%T 17,49,68,23,1,1,1,4,25,96,252,251,47,1,1,1,5,33,177,687,1406,1020,
%U 106,1,1,1,5,43,285,1537,5087,8405,4258,235,1,1,1,6,55,442,3014,14310,40546,52348,18580,551
%N Array read by antidiagonals: T(n,k) is the number of unlabeled oriented k-gonal 2-trees with n oriented polygons, n >= 0, k >= 2.
%C See section 3 of the Labelle reference.
%H Andrew Howroyd, <a href="/A340812/b340812.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 G.f. of column k: B(x) - x*B(x)^k + x*(Sum_{d|k} phi(d)*B(x^d)^(k/d))/k, where B(x) if the g.f. of column k of A340814.
%e Array begins:
%e =========================================================
%e n\k | 2 3 4 5 6 7 8 9
%e ----+----------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 1 ...
%e 1 | 1 1 1 1 1 1 1 1 ...
%e 2 | 1 1 1 1 1 1 1 1 ...
%e 3 | 2 2 3 3 4 4 5 5 ...
%e 4 | 3 7 11 17 25 33 43 55 ...
%e 5 | 6 18 49 96 177 285 442 635 ...
%e 6 | 11 68 252 687 1537 3014 5370 8901 ...
%e 7 | 23 251 1406 5087 14310 33632 70000 132533 ...
%e 8 | 47 1020 8405 40546 141582 399065 966254 2089103 ...
%e ...
%o (PARI) \\ here B(n,k) gives column k of A340814.
%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 C(n, k)={my(p=B(n,k)); Vec(p - x*p^k + x*sumdiv(k, d, eulerphi(d)*subst(p + O(x*x^(n\d)), x, x^d)^(k/d))/k)}
%o { Mat(vector(7, k, C(7, k+1)~)) }
%Y Columns 2..4 are A000055, A303742, A340813.
%Y Cf. A340811 (unoriented case), A340814 (edge-rooted case).
%K nonn,tabl
%O 0,10
%A _Andrew Howroyd_, Feb 02 2021