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

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

Offset

1,4

Comments

Also increasing involution rooted trees with n-1 nodes.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..200

O. Bodini, M. Dien, X. Fontaine, A. Genitrini, H. K. Hwang, Increasing Diamonds, in LATIN 2016: 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings Pages pp 207-219 2016, Lecture Notes in Computer Science Series Volume 9644.

C. G. Bower, Transforms (2)

Index entries for sequences related to cacti

Index entries for sequences related to rooted trees

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

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 *)

Prog

(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

Crossrefs

Cf. A001147, A091481.

Sequence in context: A189239 A074530 A159662 * A351421 A273953 A127127

Adjacent sequences: A032032 A032033 A032034 * A032036 A032037 A032038

Keyword

nonn,eigen

Author

Christian G. Bower, Apr 01 1998

Status

approved