%I #34 Nov 08 2016 03:05:01

%S 1,1,1,1,1,2,5,13,34,90,241,652,1780,4899,13581,37893,106340,299978,

%T 850187,2419788,6913658,19822439,57015620,164476023,475752469,

%U 1379553027,4009532279,11678165796,34081307147,99646051271,291845778020,856147139606,2515368741707,7400713869808,21803597196231

%N Generalized Catalan numbers: a(0) = 1, a(n) = a(n-1) + Sum_{k=1..n-4} a(k) * a(n-k).

%C a(n) = number of bargraphs of semiperimeter n-2 with no valleys of width 1 (i.e., no DHU configurations, where U=(0,1), H=(1,0), D=(0,-1)). Example: a(8) = 34 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only the one corresponding to the composition [2,1,2] has a valley of width 1. - _Emeric Deutsch_, Aug 11 2016

%F The sequence a(n-3) (for n>=3) has the g.f. 1/G(0) where G(k) = 1 - q/(1 - q - q^2 - q^3 / G(k+1) ). - _Joerg Arndt_, Dec 06 2014

%F n*a(n) +2*(-2*n+3)*a(n-1) +2*(n-3)*a(n-2) +(2*n-9)*a(n-3) +(n-6)*a(n-4) +(2*n-15)*a(n-5) +(n-9)*a(n-6)=0. - _R. J. Mathar_, May 01 2015

%F G.f.: (3+z^2+z^3-sqrt((1-2*z-3*z^2-z^3)*(1-2*z+z^2-z^3)))/2. - _Emeric Deutsch_, May 24 2016

%F The g.f. g(x) satisfies g(x) = 1+x*g(x)+(g(x)-1)*(g(x)-x^3-x^2-x-1) and 3*x^5+5*x^4+7*x^3+6*x^2+2*x-3+(-3*x^5-5*x^4-2*x^3-3*x^2-2*x+2)*g(x)+(x^6+2*x^5+x^4+2*x^3+2*x^2-4*x+1)*g'(x). - _Robert Israel_, May 25 2016

%p A023425 := proc(n)

%p option remember;

%p if n = 0 then

%p 1;

%p else

%p procname(n-1)+add(procname(k)*procname(n-k),k=1..n-4) ;

%p end if;

%p end proc: # _R. J. Mathar_, May 01 2015

%t a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-k ], {k, 1, n-4} ];

%Y Cf. A000108, A001006, A004148, A006318, A082582.

%K nonn,easy

%O 0,6

%A _Olivier Gérard_