A176610 - OEIS

%I #6 Oct 08 2016 09:15:32

%S 1,0,3,10,25,65,197,652,2203,7523,26159,92663,332747,1206641,4411883,

%T 16252550,60270497,224798517,842706069,3173330573,11998214633,

%U 45531318219,173359346313,662062569685,2535444644053,9734529981735

%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)=0, k=1 and l=1.

%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) +(2-7n)*a(n-1) +(19n-29)*a(n-2) +(110-37n)*a(n-3) +36*(n-4)*a(n-4) +12*(5-n)*a(n-5)=0. - _R. J. Mathar_, Nov 17 2011

%e a(2)=(1*0+1)+(1*0+1)+1=3. a(3)=2*1*3+2+(0^2+1)+1=10. a(4)=2*1*10+2+2*0*3+2+1=25.

%p l:=1: : k := 1 : m:=0: 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);

%K nonn

%O 0,3

%A _Richard Choulet_, Apr 21 2010