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
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
Programs
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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;
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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) = 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
Comments