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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)=0, k=-1 and l=0.
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%I #4 Mar 01 2016 16:18:45

%S 1,0,-2,-7,-18,-37,-52,10,412,1865,5740,12922,16092,-29767,-290264,

%T -1213217,-3608342,-7564363,-6023704,38816098,259037300,991747431,

%U 2756105680,5061761997,284694486,-47403203725,-254747436848

%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)=0, k=-1 and l=0.

%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=0).

%F Conjecture: (n+1)*a(n) +(-7*n+2)*a(n-1) +(19*n-29)*a(n-2) +(-17*n+40)*a(n-3) +2*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Mar 01 2016

%e a(2)=2*1*0-2=-2. a(3)=2*1*(-2)-2+0-1=-7. a(4)=2*1*(-7)-2+2*0*(-2)-2=-18.

%p l:=0: : k :=-1 : m:=0: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 : 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);

%K easy,sign

%O 0,3

%A _Richard Choulet_, Apr 27 2010