A340811 Array read by antidiagonals: T(n,k) is the number of unlabeled k-gonal 2-trees with n polygons, n >= 0, k >= 2.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 3, 5, 6, 1, 1, 1, 3, 8, 12, 11, 1, 1, 1, 4, 11, 32, 39, 23, 1, 1, 1, 4, 16, 56, 141, 136, 47, 1, 1, 1, 5, 20, 103, 359, 749, 529, 106, 1, 1, 1, 5, 26, 158, 799, 2597, 4304, 2171, 235, 1, 1, 1, 6, 32, 245, 1539, 7286, 20386, 26492, 9368, 551

Offset: 0

Author

Andrew Howroyd, Feb 02 2021

Keywords

• nonn
• tabl

Comments

See section 4 and table 1 in the Labelle reference.

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   5    8    11    16     20     26      32 ...
  5 |  6  12   32    56   103    158    245     343 ...
  6 | 11  39  141   359   799   1539   2737    4505 ...
  7 | 23 136  749  2597  7286  16970  35291   66603 ...
  8 | 47 529 4304 20386 71094 199879 483819 1045335 ...
  ...
		

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

Columns 2..12 are A000055, A054581, A094610, A094611, A094637, A094651, A094652, A094653, A094654, A094655, A094656.

Cf. A340812 (with oriented polygons).

Programs

PARI

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(p,k)={p(1) - x*p(1)^k + x*sumdiv(k, d, eulerphi(d)*p(d)^(k/d))/k}
S(p,k)={my(p2=p(2)); if(k%2, 1+x*Ser(EulerT(Vec(x*p2^(k\2) + x^2*(p2^(k-1) - p(4)^(k\2))/2 ))), my(r=p2^(k/2-1), q=1+O(x)); while(serprec(q,x)