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%I #10 Mar 16 2016 14:43:42
%S 1,4,9,35,143,648,3071,15126,76495,395086,2074699,11044027,59457897,
%T 323180520,1771081641,9774955015,54286011887,303138215322,
%U 1701016909235,9586701364893,54241695455421,307991483403216,1754468491846461
%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)=4, k=0 and l=1.
%F 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=0, l=1).
%F Conjecture: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +(-3*n+11)*a(n-2) +16*(n-3)*a(n-3) +8*(-n+4)*a(n-4)=0. - _R. J. Mathar_, Feb 29 2016
%F a(n) = Sum_{k=0..n}((C(k)*Sum_{l=0..n-k}(binomial(k+1,l)*2^l*binomial(n-2*l,n-l-k)))), where C(k) is a Catalan number (A000108). - _Vladimir Kruchinin_, Mar 15 2016
%e a(2)=2*1*4+1+9. a(3)=2*1*9+4^2+1=35.
%p l:=1: : k := 0 : m:=4: d(0):=1:d(1):=m: for n from 1 to 28 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=0,31);seq(d(n),n=0..29) ;
%o (Maxima)
%o a(n):=sum((binomial(2*k,k)/(k+1)*sum(binomial(k+1,l)*2^l*binomial(n-2*l,n-l-k),l,0,n-k)),k,0,n); /* _Vladimir Kruchinin_, Mar 15 2016 */
%Y Cf. A000108, A176604, A176605, A176606.
%K easy,nonn
%O 0,2
%A _Richard Choulet_, Apr 21 2010
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