%I #6 Mar 02 2016 15:12:37
%S 1,1,-1,-5,-17,-49,-129,-305,-609,-801,735,10911,53983,203551,651487,
%T 1796639,4084447,6188831,-4060449,-84814049,-455815457,-1824908513,
%U -6141218081,-17711864033,-42059573537,-67468774625,33608030943
%N Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=1, k=-1 and l=-1.
%F G.f f: f(z)=(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) +(-7*n+2)*a(n-1) +3*(5*n-7)*a(n-2) +(-n-10)*a(n-3) +4*(-4*n+17)*a(n-4) +8*(n-5)*a(n-5)=0. - _R. J. Mathar_, Mar 02 2016
%e a(2)=2*1*1-2-1=-1. a(3)=2*1*(-1)-2+1^2-1-1=-5. a(4)=2*1*(-5)-2+2*1*(-1)-2-1=-17.
%p l:=-1: : k := -1 : m:=1: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. A176952.
%K easy,sign
%O 0,4
%A _Richard Choulet_, Apr 29 2010