A340812 -id:A340812 - cp's OEIS frontend

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

Andrew Howroyd, Feb 02 2021

See section 4 and table 1 in the Labelle reference.

			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 ...
  ...

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.

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

Cf. A340812 (with oriented polygons).

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)

A303742 Number of unlabeled oriented edge-rooted 2-trees which have n triangles.

Original entry on oeis.org

1, 1, 1, 2, 7, 18, 68, 251, 1020, 4258, 18580, 82716, 377207, 1748250, 8227066, 39197164, 188824506, 918333933, 4504366940, 22260929867, 110763984273, 554510459987, 2791460440109, 14123739733754, 71792405634223, 366483503897357, 1878163540497162, 9660232634407657

Offset: 0

R. J. Mathar, Apr 29 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500
T. Fowler, I. Gessel, G. Labelle, P. Leroux, The specification of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, Table 1.

Column k=3 of A340812.

C(30,3) \\ See A340812 for C(n,k). - Andrew Howroyd, May 09 2021

Name clarified by Andrew Howroyd, May 09 2021

A340813 Number of oriented polygonal 2-trees with n oriented quadrilaterals.

Original entry on oeis.org

1, 1, 1, 3, 11, 49, 252, 1406, 8405, 52348, 337029, 2225333, 15002763, 102896407, 716106077, 5046649182, 35956186581, 258649116963, 1876462272125, 13717081730294, 100957850485311, 747631534534529, 5567459642741752, 41671027665802639, 313351688709886964, 2366390596263188755

Offset: 0

Andrew Howroyd, Feb 02 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.

Column k=4 of A340812.

PARI
```
C(25,4) \\ See A340812 for C(n,k).
```

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.

Original entry on oeis.org

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

Andrew Howroyd, Feb 02 2021

See section 2 of the Labelle reference.

			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 ...
  ...

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.

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

Cf. A242249, A340811, A340812.

\\ 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)))) }

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).

cp's OEIS Frontend

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

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A303742 Number of unlabeled oriented edge-rooted 2-trees which have n triangles.

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A340813 Number of oriented polygonal 2-trees with n oriented quadrilaterals.

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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.

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