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A177123 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.

%I A177123 #5 Mar 02 2016 15:27:18
%S A177123 1,7,17,87,417,2347,13497,81607,504537,3192747,20529537,133876327,
%T A177123 882924177,5879675307,39478170697,266973261127,1816729697097,
%U A177123 12431013514667,85476914070417,590327766229607,4093067887259777
%N A177123 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 A177123 To see by recurrence: a(n)=7 mod10 for n>0.
%F A177123 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 A177123 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 A177123 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 A177123 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 A177123 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 A177123 Cf. A177122.
%K A177123 easy,nonn
%O A177123 0,2
%A A177123 _Richard Choulet_, May 03 2010