A340812 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.
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, 17, 49, 68, 23, 1, 1, 1, 4, 25, 96, 252, 251, 47, 1, 1, 1, 5, 33, 177, 687, 1406, 1020, 106, 1, 1, 1, 5, 43, 285, 1537, 5087, 8405, 4258, 235, 1, 1, 1, 6, 55, 442, 3014, 14310, 40546, 52348, 18580, 551
Offset: 0
Examples
Array begins: ========================================================= n\k | 2 3 4 5 6 7 8 9 ----+---------------------------------------------------- 0 | 1 1 1 1 1 1 1 1 ... 1 | 1 1 1 1 1 1 1 1 ... 2 | 1 1 1 1 1 1 1 1 ... 3 | 2 2 3 3 4 4 5 5 ... 4 | 3 7 11 17 25 33 43 55 ... 5 | 6 18 49 96 177 285 442 635 ... 6 | 11 68 252 687 1537 3014 5370 8901 ... 7 | 23 251 1406 5087 14310 33632 70000 132533 ... 8 | 47 1020 8405 40546 141582 399065 966254 2089103 ... ...
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 column k of A340814. 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} 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)} { Mat(vector(7, k, C(7, k+1)~)) }
Formula
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.
Comments