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

0,10

Comments

See section 4 and table 1 in 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.

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   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 ...
  ...

Prog

(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)<serprec(p2, x), my(t=r*q); q=1+x*Ser(EulerT(Vec(x*t + x^2*subst(p(1)^(k-1) - t, x, x^2)/2)))); q + x*p2^(k/2-1)*(p2-q^2)/2)}
U(n, k)={my(b=B(n, k), p(d)=subst(b + O(x*x^(n\d)), x, x^d)); Vec(C(p, k) + S(p, k))/2}
{ Mat(vector(7, k, U(7, k+1)~)) }

Crossrefs

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

Cf. A340812 (with oriented polygons).

Sequence in context: A066170 A046854 A184957 * A340812 A228349 A285718

Adjacent sequences: A340808 A340809 A340810 * A340812 A340813 A340814

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 02 2021

Status

approved