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%I
%S 1,10,19,137,653,4406,27077,185856,1259601,8898900,63225681,457994141,
%T 3345121235,24706965674,183830383235,1378149812989,10393740091309,
%U 78828658428280,600737927801161,4598286755156991,35334943369372359
%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=0 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=0, l=-1).
%F Conjecture: (n+1)*a(n) +2(1-3n)*a(n-1) +(59-27n)*a(n-2) +4(18n-55)*a(n-3) +40(4-n)*a(n-4)=0. - R. J. Mathar, Nov 27 2011
%e a(2)=2*1*10-1=19. a(3)=2*1*19+100-1=137.
%p l:=-1: : k := 0 : 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;
%K easy,nonn
%O 0,2
%A _Richard Choulet_, May 04 2010
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