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A002067 a(n) = Sum_{k=0..n-1} binomial(2*n,2*k)*a(k)*a(n-k-1).
(Formerly M4458 N1889)
8
1, 1, 7, 127, 4369, 243649, 20036983, 2280356863, 343141433761, 65967241200001, 15773461423793767, 4591227123230945407, 1598351733247609852849, 655782249799531714375489, 313160404864973852338669783, 172201668512657346455126457343, 108026349476762041127839800617281 (list; graph; refs; listen; history; text; internal format)
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->infinity (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.

Markus Kuba, 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

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..inf} 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) * Integral A(x) dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015

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-Fran├žois 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

The sequence of fractions A092676/A132467 is closely related.

Periods: A122149, A122159.

Sequence in context: A274673 A215066 A092676 * A274571 A138523 A034670

Adjacent sequences:  A002064 A002065 A002066 * A002068 A002069 A002070

KEYWORD

nonn,eigen,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Alternate description, formula and comment from Christian G. Bower

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

STATUS

approved

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Last modified August 20 07:48 EDT 2019. Contains 326143 sequences. (Running on oeis4.)