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

Author

Christian G. Bower, Apr 01 1998

Keywords

• nonn
• eigen

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

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