OFFSET
0,3
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)
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
FORMULA
a(n) = b(2n+1), where e.g.f. of b satisfies B'(x)=exp(B(x)^2/2).
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]
EXAMPLE
E.g.f.: A(x) = 1 + x^2/2! + 7*x^4/4! + 127*x^6/6! + 4369*x^8/8! + ...
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,eigen,easy,nice
AUTHOR
EXTENSIONS
Alternate description, formula and comment from Christian G. Bower
New definition and more terms from Vladeta Jovovic, Oct 22 2005
STATUS
approved