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%I #8 Feb 18 2016 13:12:59
%S 1,3,7,25,95,393,1711,7741,36007,171097,826839,4050957,20074303,
%T 100438233,506677279,2574292749,13161031191,67656253081,349499197799,
%U 1813347470669,9445448148975,49375113712089,258938850241327
%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)=3, k=1 and l=-1.
%F G.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) +(7*n-5)*a(n-2) +(7*n-26)*a(n-3) +4*(-4*n+17)*a(n-4) +8*(n-5)*a(n-5)=0. - _R. J. Mathar_, Feb 18 2016
%e a(2)=2*1*3+2-1=7. a(3)=2*1*7+2+3^2+1-1=25. a(4)=2*1*25+2+2*3*7+2-1=95.
%p l:=-1: : k := 1 : m:=3: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. A176828.
%K easy,nonn
%O 0,2
%A _Richard Choulet_, Apr 27 2010
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