%I #7 Oct 08 2016 09:12:19
%S 1,10,19,136,649,4375,26893,184438,1249507,8820901,62634223,453382597,
%T 3309224059,24424411627,181601249779,1360466777260,10253089323433,
%U 77706202937131,591759519723973,4526303458541383,34756744887203401
%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)=10, 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) +(2-7n)*a(n-1) +3(17-7n)*a(n-2) +9(11n-34)*a(n-3) +108(4-n)*a(n-4) +36(n-5)*a(n-5)=0. - _R. J. Mathar_, Nov 27 2011
%e a(2)=2*1*10-2+1=19. a(3)=2*1*19-2+100-1+1=136.
%p l:=1: : k := -1 : for m from 0 to 10 do 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): od;
%Y Cf. A177200.
%K easy,nonn
%O 0,2
%A _Richard Choulet_, May 04 2010