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%I #5 Feb 21 2016 15:56:46
%S 1,5,8,38,152,743,3608,18515,96542,515525,2792780,15341492,85186412,
%T 477531833,2698428176,15355638218,87919098128,506118923897,
%U 2927616746156,17007899032118,99191713057280,580535666936861
%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)=5, 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) +(-n+11)*a(n-2) +(43*n-140)*a(n-3) +2*(-28*n+113)*a(n-4) +20*(n-5)*a(n-5)=0. - _R. J. Mathar_, Feb 21 2016
%e a(2)=2*1*5-2=8. a(3)=2*1*8-2+5^2-1=38. a(4)=2*1*38-2+2*5*8-2=152.
%p l:=0: : k := -1 : m:=5: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. A176858.
%K easy,nonn
%O 0,2
%A _Richard Choulet_, Apr 27 2010
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