%I #8 Jul 21 2021 09:35:58
%S 1,2,1,2,3,9,27,92,313,1083,3753,13063,45581,159501,559549,1967878,
%T 6937267,24511653,86797683,308003549,1095155727,3901490015,
%U 13924590847,49784694997,178293760747,639543538859,2297555097259,8265957750659
%N a(n+1) = 1 + Sum_{p=0..n} a(p)*a(n-p)+k for n>=1, with here a(0)=1, a(1)=2, k=-1 and l=-1.
%F G.f: (1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=-1, l=-1).
%F Conjecture: (n+1)*a(n) +(2-7n)*a(n-1) +(11n-13)*a(n-2) + (11n-46)*a(n-3) +4*(29-7n)*a(n-4) +12(n-5)*a(n-5)=0. - _R. J. Mathar_, Nov 21 2011
%e a(2)=2*1*2-2-1=1. a(3)=2*1*1-2+2^2-1-1=2. a(4)=2*1*2-2+2*2*1-2-1=3.
%p l:=-1: : k := -1 : m:=2:d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
%p taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 30); seq(d(n), n=0..30);
%Y Cf. A176953.
%K easy,nonn
%O 0,2
%A _Richard Choulet_, Apr 29 2010