cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Stepan Orevkov

Stepan Orevkov's wiki page.

Stepan Orevkov has authored 3 sequences.

A187014 Number of isomorphism classes of simply embedded (i.e., loop-free and without parallel edges bounding a disk) bipartite quadrangulations of RP^2 of minimum degree 3 with n nodes and n-1 faces.

Original entry on oeis.org

0, 0, 1, 2, 3, 7, 22, 57, 174, 554, 1848, 6291, 22052, 78361, 282420
Offset: 4

Author

Stepan Orevkov, Aug 30 2013

Keywords

Comments

Also the number of isomorphism classes of homologically trivial basic nets on RP^2 with n-1 nodes and n faces.
Also the number of isomorphism classes of simple quadrangulations of the sphere of minimum degree 3 with 2n nodes and 2n-2 faces which admit an orientation reversing involution without fix nodes and fix edges such that the bipartite coloring is preserved under the involution.

Crossrefs

A187013 Number of isomorphism classes of simply embedded (i.e., loop-free and without parallel edges bounding a disk) quadrangulations of RP^2 of minimum degree 3 with n nodes and n-1 faces.

Original entry on oeis.org

1, 0, 2, 3, 6, 12, 37, 95, 293, 923, 3086, 10504, 36954, 131590, 475793
Offset: 4

Author

Stepan Orevkov, Aug 30 2013

Keywords

Comments

Also the number of isomorphism classes of basic nets (polyhedra) on RP^2 with n-1 nodes and n faces.
Also the number of isomorphism classes of simple quadrangulations of the sphere of minimum degree 3 with 2n nodes and 2n-2 faces which admit an orientation reversing involution without fix nodes and fix edges.

Crossrefs

A185650 a(n) is the number of rooted trees with 2n vertices n of whom are leaves.

Original entry on oeis.org

1, 2, 8, 39, 214, 1268, 7949, 51901, 349703, 2415348, 17020341, 121939535, 885841162, 6511874216, 48359860685, 362343773669, 2736184763500, 20805175635077, 159174733727167, 1224557214545788, 9467861087020239, 73534456468877012, 573484090227222260
Offset: 1

Author

Stepan Orevkov, Aug 29 2013

Keywords

Examples

			From _Gus Wiseman_, Nov 27 2022: (Start)
The a(1) = 1 through a(3) = 8 rooted trees:
  (o)  ((oo))  (((ooo)))
       (o(o))  ((o)(oo))
               ((o(oo)))
               ((oo(o)))
               (o((oo)))
               (o(o)(o))
               (o(o(o)))
               (oo((o)))
(End)
		

Crossrefs

The ordered version is A000891, ranked by A358579.
This is the central column of A055277.
These trees are ranked by A358578.
For height = internals we have A358587.
Square trees are counted by A358589.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internals, ordered A090181.

Programs

  • Mathematica
    terms = 23;
    m = 2 terms;
    T[, ] = 0;
    Do[T[x_, z_] = z x - x + x Exp[Sum[Series[1/k T[x^k, z^k], {x, 0, j}, {z, 0, j}], {k, 1, j}]] // Normal, {j, 1, m}];
    cc = CoefficientList[#, z]& /@ CoefficientList[T[x, z] , x];
    Table[cc[[2n+1, n+1]], {n, 1, terms}] (* Jean-François Alcover, Sep 14 2018 *)
    art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
    Table[Length[Select[art[n],Count[#,{},{-2}]==n/2&]],{n,2,10,2}] (* Gus Wiseman, Nov 27 2022 *)
  • PARI
    \\ here R is A055277 as vector of polynomials
    R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
    {my(A=R(2*30)); vector(#A\2, k, polcoeff(A[2*k],k))} \\ Andrew Howroyd, May 21 2018

Extensions

Terms a(20) and beyond from Andrew Howroyd, May 21 2018