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

0,6

Comments

See section 2 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

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).

Example

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 ...
  ...

Prog

(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)))) }

Crossrefs

Columns k=2..6 are A000081(n+1), A005750(n+1), A052751, A052773, A052781.

Cf. A242249, A340811, A340812.

Sequence in context: A122175 A073165 A137153 * A063841 A256161 A137596

Adjacent sequences: A340811 A340812 A340813 * A340815 A340816 A340817

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 02 2021

Status

approved