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.

A361245 Number of noncrossing 2,3 cacti with n nodes.

Original entry on oeis.org

1, 1, 1, 4, 20, 115, 715, 4683, 31824, 222300, 1586310, 11514030, 84742320, 630946446, 4743789260, 35965715780, 274659794160, 2110810059795, 16312695488265, 126693445737170, 988340783454380, 7740875273884445, 60846920004855985, 479854293574853085
Offset: 0

Views

Author

Andrew Howroyd, Mar 08 2023

Keywords

Comments

A 2,3 cactus is a cactus composed of bridges and triangles.

Links

Crossrefs

Cf. A091481, A091485, A091486, A091487, A361242, A361244.

Programs

  • PARI
    seq(n) = Vec(1 + x/(1 - serreverse((sqrt(1 + 4*x + O(x^n)) - 1)*(1 - x)^2/2)))

A091485 Number of labeled 2,3 cacti (triangular cacti with bridges).

Original entry on oeis.org

1, 1, 4, 28, 290, 3996, 68992, 1434112, 34895772, 973450000, 30636233936, 1074020373504, 41510792057176, 1753764940408768, 80412829785000000, 3977094146761424896, 211058327532167398928, 11963018212810373415168, 721321146876339731628352
Offset: 1

Views

Author

Christian G. Bower, Jan 14 2004

Keywords

Comments

As Alois P. Heinz has pointed out, the e.g.f in the Example section does not match the offset. However, the identity a(n) = A091481(n)/n holds with the present offset of 1. - N. J. A. Sloane, Jun 23 2017

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 290*x^4/4! + 3996*x^5/5! +...

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.84).

Links

Crossrefs

Cf. A091481, A091487, A000085.

Programs

  • Mathematica
    CoefficientList[1/x InverseSeries[x/Exp[x+x^2/2]+O[x]^20], x] Range[0, 18]! (* Jean-François Alcover, Aug 06 2018 *)

Formula

a(n) = A091481(n)/n.
From Paul D. Hanna, Jun 01 2012: (Start)
E.g.f.: (1/x)*Series_Reversion( x/exp(x+x^2/2) ).
E.g.f. satisfies: A(x) = exp( x*A(x) + x^2*A(x)^2/2 ).
E.g.f. satisfies: A( x/exp(x+x^2/2) ) = exp(x+x^2/2).
(End)
a(n+1) = n! * Sum_{k=0..n} (1/2)^(n-k) * (n+1)^(k-1) * binomial(k,n-k)/k!. - Seiichi Manyama, Aug 19 2023

A032035 Number of increasing rooted 2,3 cacti (triangular cacti with bridges) with n-1 nodes.

Original entry on oeis.org

1, 1, 1, 3, 13, 77, 573, 5143, 54025, 650121, 8817001, 133049339, 2210979381, 40118485237, 789221836741, 16730904387183, 380227386482641, 9221550336940241, 237724953543108753, 6491255423787076915, 187156557809878784797, 5681772224922980536413
Offset: 1

Views

Author

Christian G. Bower, Apr 01 1998

Keywords

Comments

Also increasing involution rooted trees with n-1 nodes.

Links

Crossrefs

Cf. A001147, A091481.

Programs

  • Maple
    A:= proc(n) option remember; if n=0 then x else convert(series(Int(exp(A(n-1)+ A(n-1)^2/2), x), x=0, n+1), polynom) fi end; a:= n-> if n=1 then 1 else coeff(A(n-1), x, n-1)*(n-1)! fi: seq(a(n), n=1..20); # Alois P. Heinz, Aug 22 2008
  • Mathematica
    CoefficientList[Series[Sqrt[2]*InverseErf[Sqrt[2/(E*Pi)] x + Erf[1/Sqrt[2]]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 07 2014 *)
m = 22; A[_] = 0;
Do[A[x_] = Integrate[Exp[A[x] + A[x]^2/2], x] + O[x]^m, {m}];
CoefficientList[1 + A[x], x]*Range[0, m-1]! (* Jean-François Alcover, Sep 29 2019 *)
  • PARI
    seq(n)={my(p=x+O(x*x^(n%2))); for(i=1, n\2, p=intformal(1 + intformal(1/(1-p)))); Vec(serlaplace(p))} \\ Andrew Howroyd, Sep 19 2018

Formula

E.g.f. of a(n+1) satisfies A'(x) = exp(A(x)+A(x)^2/2).
E.g.f. satisfies A''(x) = 1/(1-A(x)).
Shifts left 2 places under "AIJ" (ordered, indistinct, labeled) transform.
a(1) = a(2) = 1; a(n) = Sum_{k=1..n-2} binomial(n-2,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020
Showing 1-3 of 3 results.

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

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