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

%I A002067 M4458 N1889 #102 Jul 07 2024 12:10:53
%S A002067 1,1,7,127,4369,243649,20036983,2280356863,343141433761,
%T A002067 65967241200001,15773461423793767,4591227123230945407,
%U A002067 1598351733247609852849,655782249799531714375489,313160404864973852338669783,172201668512657346455126457343,108026349476762041127839800617281
%N A002067 a(n) = Sum_{k=0..n-1} binomial(2*n,2*k)*a(k)*a(n-k-1).
%C A002067 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.)
%C A002067 a(n) is (2n)!/2^n times the n-th coefficient in the series for inverf(2x/sqrt(Pi)). - _Paul Barry_, Apr 12 2010
%C A002067 Number of ordered bilabeled increasing trees with 2n labels. - _Markus Kuba_, Nov 17 2014
%C A002067 Limit_{n->oo} (a(n)/(n!)^2)^(1/n) = 8/Pi. - _Vaclav Kotesovec_, Nov 19 2014
%C A002067 From _David Callan_, Jul 21 2017: (Start)
%C A002067 Conjectures:
%C A002067 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.
%C A002067 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)
%D A002067 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, cf. Chapter 5.
%D A002067 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002067 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002067 T. D. Noe, <a href="/A002067/b002067.txt">Table of n, a(n) for n = 0..50</a>
%H A002067 L. Carlitz, <a href="http://projecteuclid.org/euclid.pjm/1103035736">The inverse of the error function</a>, Pacific J. Math., 13 (1963), 459-470. [See Eq. 1.3 and Section 6.]
%H A002067 D. Dominici, <a href="http://arxiv.org/abs/math/0607230">Asymptotic analysis of the derivatives of the inverse error function</a>, arXiv:math/0607230 [math.CA], 2006-2007.
%H A002067 A. J. E. M. Janssen, <a href="https://www.eurandom.tue.nl/wp-content/uploads/2023/03/mrt2023_final.pdf">Analysis of a constrained initial value for an ODE arising in the study of a power-flow model</a>, Eindhoven Univ. Tech. (Netherlands, 2023).
%H A002067 Markus Kuba and Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], 2014.
%H A002067 Wikipedia, <a href="http://en.wikipedia.org/wiki/Error_function">Error Function</a>
%H A002067 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%F A002067 a(n) = b(2n+1), where e.g.f. of b satisfies B'(x)=exp(B(x)^2/2).
%F A002067 a(n) = (2n)! * A092676(n) / (2^n*A092677(n)). - _Paul Barry_, Apr 12 2010
%F A002067 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
%F A002067 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
%F A002067 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_]
%e A002067 E.g.f.: A(x) = 1 + x^2/2! + 7*x^4/4! + 127*x^6/6! + 4369*x^8/8! + ...
%p A002067 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;
%t A002067 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_ *)
%o A002067 (PARI) /* E.g.f. A(x) = exp( Integral A(x) * Integral A(x) dx dx ): */
%o A002067 {a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A * intformal( A + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
%o A002067 for(n=0,20,print1(a(2*n),", ")) \\ _Paul D. Hanna_, Jun 02 2015
%o A002067 (PARI) /* By definition: */
%o A002067 {a(n) = if(n==0,1,sum(k=0,n-1, binomial(2*n,2*k)*a(k)*a(n-k-1)))}
%o A002067 for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Jun 02 2015
%Y A002067 The sequence of fractions A092676/A132467 is closely related.
%Y A002067 Periods: A122149, A122159.
%K A002067 nonn,eigen,easy,nice
%O A002067 0,3
%A A002067 _N. J. A. Sloane_
%E A002067 Alternate description, formula and comment from _Christian G. Bower_
%E A002067 New definition and more terms from _Vladeta Jovovic_, Oct 22 2005