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%I #6 Feb 18 2016 13:13:25
%S 1,5,11,49,211,1037,5267,27953,152075,845709,4780923,27402033,
%T 158842179,929655949,5485858531,32603081969,194973609467,
%U 1172405681165,7084340575307,42994921155441,261963852143283,1601804565028621
%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=-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) +(-n+11)*a(n-2) +(31*n-98)*a(n-3) +4*(-10*n+41)*a(n-4) +16*(n-5)*a(n-5)=0. - _R. J. Mathar_, Feb 18 2016
%e a(2)=2*1*5+2-1=11. a(3)=2*1*11+2+5^2+1-1=49. a(4)=2*1*49+2+2*5*11+2-1=211.
%p l:=-1: : 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 : 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. A176830.
%K easy,nonn
%O 0,2
%A _Richard Choulet_, Apr 27 2010
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