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.

Showing 1-3 of 3 results.

A214337 Triangle read by rows: T(n,k) = number of rooted maps with n vertices and k faces on a non-orientable surface of type 3/2 (0 <= k <= n).

Original entry on oeis.org

0, 0, 41, 0, 690, 16925, 0, 7150, 237652, 4306778, 0, 58760, 2518957, 56864524, 910734615, 0, 420182, 22417804, 613687758, 11675167470, 174833737848
Offset: 0

Views

Author

N. J. A. Sloane, Jul 27 2012

Keywords

Examples

			Triangle begins:
  0;
  0,     41;
  0,    690,    16925;
  0,   7150,   237652,   4306778;
  0,  58760,  2518957,  56864524,   910734615;
  0, 420182, 22417804, 613687758, 11675167470, 174833737848;
  ...
		

Crossrefs

Diagonals give A118448, A214335, A213336, A213338.
Cf. A214806.

A118449 Number of rooted n-edge one-vertex maps on a non-orientable genus-4 surface (dually: one-face maps).

Original entry on oeis.org

0, 488, 11660, 160680, 1678880, 14771680, 115457832, 827303280, 5545466520, 35257287120, 214730922120, 1262004908528, 7197437563680, 40007524376960, 217501266966160, 1159737346931040, 6079078540464072, 31385516059734960
Offset: 3

Views

Author

Valery A. Liskovets, May 04 2006

Keywords

Comments

One-vertex maps on a non-orientable genus-3 surface are counted by A118448. Such maps are also called bouquets of loops (and their duals are called unicellular maps).

References

  • E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
  • D. M. Jackson and T. I. Visentin, An atlas of the smaller maps in orientable and nonorientable surfaces. CRC Press, Boca Raton, 2001.

Crossrefs

Cf. A118448. A diagonal of A214806.

Programs

  • Mathematica
    With[{r=Sqrt[1-4x]},Drop[CoefficientList[Series[-(r-1)^4 (r+1)^3 (65r^3+ 337r^2- 433r-945)/(256r^11),{x,0,20}],x],3]] (* Harvey P. Dale, Aug 05 2019 *)

Formula

O.g.f.: -(R-1)^4(R+1)^3(65R^3+337R^2-433R-945)/(256R^11), where R=sqrt(1-4x).
a(n) ~ n^(9/2) * 2^(2*n-3) / sqrt(Pi) * (1 - 2*sqrt(Pi)/(3*sqrt(n))). - Vaclav Kotesovec, Oct 27 2024

A118447 Number of rooted n-edge one-vertex maps on the Klein bottle (dually: one-face maps).

Original entry on oeis.org

4, 42, 304, 1870, 10488, 55412, 280768, 1379286, 6616360, 31144300, 144367584, 660746892, 2991902704, 13424189160, 59758420736, 264191654758, 1160934273288, 5074150057916, 22071747625120, 95596117130724
Offset: 2

Views

Author

Valery A. Liskovets, May 04 2006

Keywords

Comments

One-vertex maps on the projective plane are counted by A000346 and one-vertex maps on a non-orientable genus-3 surface by A118448. Such maps are also called bouquets of loops (and their duals are called unicellular maps).

References

  • E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
  • D. M. Jackson and T. I. Visentin, An atlas of the smaller maps in orientable and nonorientable surfaces. CRC Press, Boca Raton, 2001.

Programs

  • Mathematica
    ((R - 1)^2 (R + 1) (R + 3)/(8 R^5) /. R -> Sqrt[1 - 4x]) + O[x]^22 // CoefficientList[#, x]& // Drop[#, 2]& (* Jean-François Alcover, Aug 28 2019 *)

Formula

O.g.f.: (R-1)^2(R+1)(R+3)/8R^5, where R=sqrt(1-4x).
Conjecture: -(n-2)*(n-1)^2*a(n) +2*n*(4*n-5)*(n-2)*a(n-1) -8*n*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jun 22 2016
a(n) ~ n^(3/2) * 2^(2*n-1) / sqrt(Pi) * (1 - sqrt(Pi/n)/2). - Vaclav Kotesovec, Aug 28 2019
Showing 1-3 of 3 results.