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.

A035082 Number of rooted polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 5, 13, 27, 67, 157, 390, 963, 2437, 6186, 15908, 41127, 107148, 280569, 738675, 1953054, 5185364, 13816018, 36934431, 99030038, 266254593, 717652816, 1938831589, 5249221790, 14240130827, 38702218134, 105367669062
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures.
  • F. Harary and E. M. Palmer, Graphical Enumeration, p. 71

Links

Crossrefs

Cf. A003080, A035083, A035084, A035085, A035086, A035087, A035088.

Programs

  • PARI
    BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); concat([0], Vec(p))} \\ Andrew Howroyd, Aug 30 2018

Formula

Shifts left under transform T where Ta = EULER(BIK(a)-a).

A035086 Number of increasing rooted polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

1, 0, 1, 3, 19, 135, 1204, 12537, 150556, 2043930, 30969211, 517973148, 9478800604, 188381470095, 4040440921699, 93020386382742, 2287969523647171, 59877222907995675, 1661259526266784171, 48705364034046758493, 1504614657169716311674, 48848750173492332588525
Offset: 1

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Comments

Nodes are numbered and the numbers increase as you move away from the root to any point on the same polygon.

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301 and Chapter 5.
  • F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.

Links

Crossrefs

Cf. A035082-A035088.

Programs

  • Maple
    A:= proc(n) option remember; if n<=1 then x else convert(series(Int(exp(A(n-1)^2/ (2-2*A(n-1))), x), x=0, n+1), polynom) fi end; a:= n-> coeff(A(n), x, n)*n!: seq(a(n), n=1..22); # Alois P. Heinz, Aug 22 2008
  • Mathematica
    max = 22; sy = Series[Integrate[E^(-(y^2/(2-2*y))), y], {y, 0, max}]; sx = Normal[ InverseSeries[sy, x]]; a[n_] := Coefficient[sx, x, n]*n!; Table[a[n], {n, 1, max }] (* Jean-François Alcover, Feb 24 2015 *)

Formula

E.g.f. satisfies A'(x) = exp(A(x)^2/(2-2*A(x))).

A035087 Number of labeled rooted polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

1, 0, 3, 12, 135, 1440, 20895, 342720, 6585705, 142430400, 3449279295, 92207808000, 2699909867655, 85900402748160, 2951318065570875, 108894519775641600, 4294542443185019025, 180277244225580902400, 8025792422657714379675, 377695544010698833920000
Offset: 1

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
  • Harary and E. M. Palmer, Graphical Enumeration, p. 71
  • F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141
  • F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953

Links

Crossrefs

Cf. A035082-A035088.

Programs

  • Maple
    A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)^2/ (2-2*A(n-1))), x=0, n+1), polynom) fi end: a:= n-> coeff(A(n), x, n)*n!: seq(a(n), n=1..30); # Alois P. Heinz, Aug 22 2008
  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[E^(x^2/(2*(x-1)))*x,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

Formula

E.g.f. satisfies A(x)=x*exp(A(x)^2/(2-2*A(x))).
a(n) ~ (1-s)^2 * sqrt(2/(6-11*s+4*s^2)) * n^(n-1) / (s * exp(1 - s^2/(2*(1-s))))^n, where s = 0.5391888728108891165... is the root of the equation 2-4*s+s^3=0. - Vaclav Kotesovec, Jan 08 2014
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.