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

0,10

Comments

See section 3 of the Labelle reference.

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.

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.

Example

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 ...
  ...

Prog

(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)~)) }

Crossrefs

Columns 2..4 are A000055, A303742, A340813.

Cf. A340811 (unoriented case), A340814 (edge-rooted case).

Sequence in context: A046854 A184957 A340811 * A228349 A285718 A205792

Adjacent sequences: A340809 A340810 A340811 * A340813 A340814 A340815

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 02 2021

Status

approved