A002067 a(n) = Sum_{k=0..n-1} binomial(2*n,2*k)*a(k)*a(n-k-1).

1, 1, 7, 127, 4369, 243649, 20036983, 2280356863, 343141433761, 65967241200001, 15773461423793767, 4591227123230945407, 1598351733247609852849, 655782249799531714375489, 313160404864973852338669783, 172201668512657346455126457343, 108026349476762041127839800617281

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• eigen
• easy
• nice

Comments

Number of increasing rooted triangular cacti of 2n+1 nodes. (In an increasing rooted graph, nodes are numbered and numbers increase as you move away from the root.)

a(n) is (2n)!/2^n times the n-th coefficient in the series for inverf(2x/sqrt(Pi)). - Paul Barry, Apr 12 2010

Number of ordered bilabeled increasing trees with 2n labels. - Markus Kuba, Nov 17 2014

Limit_{n->oo} (a(n)/(n!)^2)^(1/n) = 8/Pi. - Vaclav Kotesovec, Nov 19 2014

From David Callan, Jul 21 2017: (Start)

Conjectures:

a(n) is the Hafnian of the triangular array (u(i,j))_{1 <= i < j <= 2n} with u(i,j)=i. The Hafnian is the same as the Pfaffian except without the alternating signs just as the permanent of a matrix is the determinant without the signs.

a(n) is the total weight of Dyck n-paths with weight defined as follows. Given a Dyck path, for each upstep, record its position in the path and the height of its upper endpoint; then multiply together all of these positions and heights. For example, the Dyck 4-path P = UUDUUDDD has upsteps in positions 1,2,4,5 ending at heights 1,2,2,3 respectively, and hence weight(P) = 480. (In fact the positions determine the heights because, for the k-th upstep, position + height = 2k.) (End)

Examples

			E.g.f.: A(x) = 1 + x^2/2! + 7*x^4/4! + 127*x^6/6! + 4369*x^8/8! + ...
		

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, cf. Chapter 5.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..50
• L. Carlitz, The inverse of the error function, Pacific J. Math., 13 (1963), 459-470. [See Eq. 1.3 and Section 6.]
• D. Dominici, Asymptotic analysis of the derivatives of the inverse error function, arXiv:math/0607230 [math.CA], 2006-2007.
• A. J. E. M. Janssen, Analysis of a constrained initial value for an ODE arising in the study of a power-flow model, Eindhoven Univ. Tech. (Netherlands, 2023).
• Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.
• Wikipedia, Error Function
• Index entries for sequences related to cacti

Crossrefs

The sequence of fractions A092676/A132467 is closely related.

Periods: A122149, A122159.

Programs

Maple

a:=proc(n) option remember; if n <= 0 then RETURN(1); else RETURN( add( binomial(2*n,2*k)*a(k)*a(n-k-1), k=0..n-1 ) ); fi; end;

Mathematica

max = 16; se = Series[ InverseErf[ 2*x/Sqrt[Pi] ], {x, 0, 2*max+1} ]; a[n_] := (2n+1)!/2^n*Coefficient[ se, x, 2*n+1]; Table[ a[n], {n, 0, max} ] (* Jean-François Alcover, Mar 07 2012, after Paul Barry *)

PARI

/* E.g.f. A(x) = exp( Integral A(x) * Integral A(x) dx dx ): */
{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A * intformal( A + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", ")) \\ Paul D. Hanna, Jun 02 2015

PARI

/* By definition: */
{a(n) = if(n==0,1,sum(k=0,n-1, binomial(2*n,2*k)*a(k)*a(n-k-1)))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Jun 02 2015

Formula

a(n) = b(2n+1), where e.g.f. of b satisfies B'(x)=exp(B(x)^2/2).

a(n) = (2n)! * A092676(n) / (2^n*A092677(n)). - Paul Barry, Apr 12 2010

a(n) = 1/2^n * A026944(n+1). Let D denote the operator g(x) -> (1/sqrt(2))*d/dx(exp(x^2)*g(x)). Then a(n) = D^(2*n)(1) evaluated at x = 0. - Peter Bala, Sep 08 2011

E.g.f. B(x)=Sum_{n>=1} a(n-1)*x^(2*n)/(2*n)! satisfies differential equation B''(x) - B(x)*B''(x) - 1 = 0, B'(0)=1/2. - Vladimir Kruchinin, Aug 12 2019

E.g.f. satisfies: A(x) = exp( Integral A(x)*B(x) dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and B(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)!, and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015 [formula revised by Paul D. Hanna, Jul 06 2024 following a suggestion from Petros Hadjicostas]

Extensions

Alternate description, formula and comment from Christian G. Bower

New definition and more terms from Vladeta Jovovic, Oct 22 2005