%I #5 Mar 02 2016 15:27:18

%S 1,7,17,87,417,2347,13497,81607,504537,3192747,20529537,133876327,

%T 882924177,5879675307,39478170697,266973261127,1816729697097,

%U 12431013514667,85476914070417,590327766229607,4093067887259777

%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)=7, k=1 and l=1.

%C To see by recurrence: a(n)=7 mod10 for n>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=1).

%F Conjecture: +(n+1)*a(n) +(-7*n+2)*a(n-1) +9*(-n+3)*a(n-2) +(47*n-142)*a(n-3) +48*(-n+4)*a(n-4) +16*(n-5)*a(n-5)=0. - _R. J. Mathar_, Mar 02 2016

%e a(2)=2*1*7+2+1=17. a(3)=2*1*17+2+7^2+1+1=87. a(4)=2*1*87+2+2*7*17+2+1=417.

%p l:=1: : k := 1 : m :=7: d(0):=1:d(1):=m: for n from 1 to 32 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, 34); seq(d(n), n=0..32);

%Y Cf. A177122.

%K easy,nonn

%O 0,2

%A _Richard Choulet_, May 03 2010