|
%I #4 Mar 01 2016 16:15:29
%S 1,1,5,15,45,151,549,2083,8133,32487,132141,545299,2277021,9603111,
%T 40844629,174997363,754562037,3271847975,14257744125,62407576979,
%U 274256671949,1209604653095,5352444701861,23755193862131
%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)=1, 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) +(-7*n+2)*a(n-1) +3*(5*n-7)*a(n-2) +(-25*n+74)*a(n-3) +24*(n-4)*a(n-4) +8*(-n+5)*a(n-5)=0. - _R. J. Mathar_, Mar 01 2016
%e a(2)=2*1*1+2+1=5. a(3)=2*1*5+2+1^2+1+1=15. a(4)=2*1*15+2+2*1*5+2+1=45.
%p l:=1: : k := 1 : m :=1: 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 easy,nonn
%O 0,3
%A _Richard Choulet_, Apr 21 2010
|
