Noemie Combe - cp's OEIS frontend

A283103 Number of A'Campo forests of degree n and co-dimension 5.

0, 0, 0, 4, 1380, 75600, 2340744, 54275296, 1055436228, 18230184752, 289150871152, 4300858168200, 60843411796440

Offset: 1

Noemie Combe, Feb 28 2017

nonn

a(n) is the number of A'Campo forests of degree n and of co-dimension 5.

			For n<4, the number of A'Campo forests of degree n  and co-dimension 5 is zero.
For n = 4 the number of A'Campo forests of co-dimension 5 is 4.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press (2009).

N. Combe, V. Jugé, Counting bi-colored A'Campo forests, arXiv:1702.07672 [math.AG], 2017.

Cf. A283101, A283102, A283049, A277877.

a(n) is obtained by using the generating function N_{1} =1+yN_{2}^4 and (1-N_{2} +2yN_{2}^4 -yN_{2}^{5} +xyN_{2}^{6} +y^{2}N_{2}^{8})(1+yN_{2}^{4}-xyN_{2}^{5})+x^3y^{2}N_{2}^{9} =0, where N_{1}(x,y)=\sum_{n}N_{1}'(5,n)x^{5}y^{n} and N_{1}'(5,n) is the number of A'Campo forests with co-dimension 5; N_{2}(x,y)=\sum_{n}N_{2}'(5,n)x^{5}y^{n} where N_{2}'(5,n) is the number of partial configurations.

Added crossrefs

A283102 Number of A'Campo forests of degree n and co-dimension 4.

Original entry on oeis.org

0, 0, 0, 80, 4845, 138792, 2893338, 50507680, 787265325, 11345154600, 154362306956, 2010147294672, 25288375607950

Offset: 1

Noemie Combe, Feb 28 2017

We can prove this using generating functions.

			For n=1, n=2 and n=3, the number of A'Campo forests of co-dimension 4 is zero.
For n=4 the number of A'Campo forests of co-dimension 4 is 80.

P. Flajolet R. Sedgewick, Analytic Combinatorics, Cambridge University Press (2009)

N. Combe, V. Jugé, Counting bi-colored A'Campo forests arXiv:1702.07672 [Math.AG], 2017.

Cf. A283049, A277877, A283101, A283102, A283103.

a(n) is obtained by using the generating function N_{1} =1+yN_{2}^4 and (1-N_{2} +2yN_{2}^4 -yN_{2}^{5} +xyN_{2}^{6} +y^{2}N_{2}^{8})(1+yN_{2}^{4}-xyN_{2}^{5})+x^3y^{2}N_{2}^{9} =0, where N_{1}(x,y)=\sum_{n}N_{1}'(4,n)x^{4}y^{n} and N_{1}'(4,n) is the number of A'Campo forests with co-dimension 4; N_{2}(x,y)=\sum_{n}N_{2}'(4,n)x^{4}y^{n} where N_{2}'(4,n) is the number of partial configurations.

A283101 Numbers of A'Campo forests of degree n>2 and co-dimension 3.

Original entry on oeis.org

0, 0, 4, 344, 8760, 157504, 2388204, 32737984, 419969088, 5141235840, 60795581132, 700024311536, 7892352548080

Offset: 1

Noemie Combe, Feb 28 2017

nonn

We can prove this using generating functions.

			For n=3, there exist four A'Campo forests of co-dimension 3 and degree 3.
For n=2 there do not exist any A'Campo forests of co-dimension 3.

P. Flajolet R. Sedgewick, Analytic Combinatorics, Cambridge University Press (2009)

N. Combe, V. Jugé, Counting bi-colored A'Campo forests, arXiv:1702.07672 [Math.AG], 2017.

a(n) is obtained by using the generating function N_{1} =1+yN_{2}^4 and (1-N_{2} +2yN_{2}^4 -yN_{2}^{5} +xyN_{2}^{6} +y^{2}N_{2}^{8})(1+yN_{2}^{4}-xyN_{2}^{5})+x^3y^{2}N_{2}^{9} =0, where N_{1}(x,y)=\sum_{n}N_{1}'(3,n)x^{3}y^{n} and N_{1}'(3,n) is the number of A'Campo forests with co-dimension 3; N_{3}(x,y)=\sum_{n}N_{3}'(3,n)x^{3}y^{n} where N_{3}'(3,n) is the number of partial configurations.

A283049 Numbers of configurations of A'Campo forests with co-dimension 1 and degree n>0.

Original entry on oeis.org

0, 4, 48, 480, 4560, 42504, 393120, 3624768, 33390720, 307618740, 2835722032, 26162863584, 241614915360, 2233533229200, 20667453710400, 191422799835264, 1774573628661504, 16465220088660432, 152894968403313600, 1420856831349155200, 13213537097286612240

Offset: 0

Noemie Combe, Feb 27 2017

nonn

We can prove this using generating functions. a(n) is given also by 4*binomial(4n,n-2), for n>1.

			For n=2 the a(2)=4 solutions are the number of A'Campo forests with co-dimension 1 and degree 2.

Indranil Ghosh, Table of n, a(n) for n = 0..500
N. Combe, V. Jugé, Counting bi-colored A'Campo forests, arXiv:1702.07672 [math.AG], 2017.

Table[4*Binomial[4n,n-2],{n,1,23}] (* Indranil Ghosh, Feb 28 2017 *)

a(n) = 4*binomial(4*n,n-2) \\ Indranil Ghosh, Feb 28 2017

a(n) = 4*binomial(4n,n-2), for n>1.

A277877 Number of A'Campo forests of degree n>1 and co-dimension 2.

Original entry on oeis.org

0, 30, 608, 8740, 109296, 1269450, 14096320, 151927776, 1603346160, 16659866938, 171064877280

Offset: 1

Noemie Combe, Feb 27 2017

We can prove this using generating functions.

			For n=3 we have a(3)=30 A'Campo forests of co-dimension 2.

P. Flajolet R. Sedgewick, Analytic Combinatorics, Cambridge University Press (2009)

N. Combe, V. Jugé, Counting bi-colored A'Campo forests, arXiv:1702.07672 [math.AG], 2017.

a(n) is obtained by using the generating function N_{1} =1+yN_{2}^4 and (1-N_{2} +2yN_{2}^4 -yN_{2}^{5} +xyN_{2}^{6} +y^{2}N_{2}^{8})(1+yN_{2}^{4}-xyN_{2}^{5})+x^3y^{2}N_{2}^{9} =0, where N_{1}(x,y)=\sum_{n}N_{1}'(2,n)x^{2}y^{n} and N_{1}'(2,n) is the number of A'Campo forests with co-dimension 2; N_{2}(x,y)=\sum_{n}N_{2}'(2,n)x^{2}y^{n} where N_{2}'(2,n) is the number of partial configurations.

cp's OEIS Frontend

User: Noemie Combe

A283103 Number of A'Campo forests of degree n and co-dimension 5.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A283102 Number of A'Campo forests of degree n and co-dimension 4.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A283101 Numbers of A'Campo forests of degree n>2 and co-dimension 3.

Views

Author

Keywords

Comments

Examples

References

Links

Formula

A283049 Numbers of configurations of A'Campo forests with co-dimension 1 and degree n>0.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

PARI

Formula

A277877 Number of A'Campo forests of degree n>1 and co-dimension 2.

Views

Author

Keywords

Comments

Examples

References

Links

Formula