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A176678 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)=2, k=0 and l=-1.

%I A176678 #4 Feb 18 2016 14:30:13
%S A176678 1,2,3,9,29,102,373,1408,5441,21428,85697,347133,1421315,5872986,
%T A176678 24459731,102570877,432725309,1835333352,7821313273,33472882591,
%U A176678 143804772471,619960227498,2681200476223,11629248891246,50574022963079
%N A176678 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)=2, k=0 and l=-1.
%F A176678 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 A176678 Conjecture: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +5*(n-1)*a(n-2) +4*(2*n-7)*a(n-3) +8*(-n+4)*a(n-4)=0. - _R. J. Mathar_, Feb 18 2016
%e A176678 a(2)=2*1*2-1=3. a(3)=2*1*3+2^2-1=9. a(4)=2*1*9+2*2*3-1=29.
%p A176678 l:=-1: : k := 0 : m:=2: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 A176678 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 A176678 Cf. A176675, A176677.
%K A176678 nonn
%O A176678 0,2
%A A176678 _Richard Choulet_, Apr 23 2010