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A176830 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)=4, k=1 and l=-1.

%I A176830 #11 Feb 18 2016 13:13:13
%S A176830 1,4,9,36,147,667,3163,15610,79121,409697,2157185,11514865,62167261,
%T A176830 338870851,1862411885,10308811304,57417591755,321563019095,
%U A176830 1809699812171,10229230730987,58047787492463,330576576766193
%N A176830 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)=4, k=1 and l=-1.
%F A176830 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 A176830 Conjecture: (n+1)*a(n) +(-7*n+2)*a(n-1) +3*(n+1)*a(n-2) +(19*n-62)*a(n-3) +4*(-7*n+29)*a(n-4) +12*(n-5)*a(n-5)=0. - _R. J. Mathar_, Feb 18 2016
%e A176830 a(2)=2*1*4+2-1=9. a(3)=2*1*9+2+4^2+1-1=36. a(4)=2*1*36+2+2*4*9+2-1=147.
%p A176830 l:=-1: : k := 1 : m:=4: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);
%t A176830 CoefficientList[Series[(1+Sqrt[1-x (6+x (3+4 x (-4+3 x)))]/(-1+x))/ (2 x),{x,0,30}],x] (* _Harvey P. Dale_, May 11 2011 *)
%Y A176830 Cf. A176829.
%K A176830 easy,nonn
%O A176830 0,2
%A A176830 _Richard Choulet_, Apr 27 2010