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.
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, 1, 7, 46, 229, 647, 702, 115, 1, 1, 8, 64, 421, 1832, 4167, 3177, 286, 1, 1, 9, 85, 699, 4191, 15583, 27847, 14830, 719, 1, 1, 10, 109, 1079, 8325, 44322, 137791, 191747, 70678, 1842
Offset: 0
Examples
Array begins: ============================================================ n\k | 2 3 4 5 6 7 8 ----+------------------------------------------------------- 0 | 1 1 1 1 1 1 1 ... 1 | 1 1 1 1 1 1 1 ... 2 | 2 3 4 5 6 7 8 ... 3 | 4 10 19 31 46 64 85 ... 4 | 9 39 107 229 421 699 1079 ... 5 | 20 160 647 1832 4191 8325 14960 ... 6 | 48 702 4167 15583 44322 105284 220193 ... 7 | 115 3177 27847 137791 487662 1385888 3374267 ... 8 | 286 14830 191747 1255202 5527722 18795035 53275581 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325
- G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.
Crossrefs
Programs
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PARI
\\ here B(n,k) gives g.f. of k-th column. EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} B(n, k)={my(p=1+O(x)); for(n=1, n, p=1+x*Ser(EulerT(Vec(p^(k-1))))); p} { Mat(vector(7, k, Col(B(7, k+1)))) }
Formula
Column k is the Euler transform of column k+1 of A242249.
G.f. of column k: A(x) satisfies A(x) = exp(Sum_{i>0} x^i*A(x^i)^(k-1)/i).
Comments