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A340814 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.

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