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%I #25 Jun 07 2023 04:36:37
%S 1,1,3,10,35,132,519,2105,8746,37033,159229,693343,3051290,13550083,
%T 60642857,273248824,1238567263,5643738611,25837579578,118785766683,
%U 548182891007,2538522337214,11792272546723,54936210525388
%N G.f. A(x) satisfies A(x) = 1 + x*(1+x+x^2)*A(x)^2.
%H G. C. Greubel, <a href="/A084781/b084781.txt">Table of n, a(n) for n = 0..1000</a>
%F a(0)=1; for n > 0, a(n) = Sum_{j=n-3..n-1} Sum_{i=0..j} a(i)*a(j-i). - Mario Catalani (mario.catalani(AT)unito.it), Jun 19 2003
%F G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x+x^2+x^3 (continued fraction); equivalently g.f. C(x+x^2+x^3) where C(x) is the g.f. for the Catalan numbers (A000108). - _Joerg Arndt_, Mar 18 2011
%F a(n) = Sum_{m=1..n} (Sum_{k=m..n} ((Sum_{j=0..k} binomial(j,n-3*k+2*j) * binomial(k,j))) * binomial(-m+2*k-1,k-1))/k))*m). - _Vladimir Kruchinin_, May 28 2011
%F Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + (7*n-11)*a(n-2) + 12*(n-2)*a(n-3) + 2*(4*n-11)*a(n-4) + 2*(2*n-7)*a(n-5). - _Vaclav Kotesovec_, Sep 17 2013
%F a(n) ~ 1/sqrt(3)*sqrt(-(1350 + 66*sqrt(131)*sqrt(3))^(2/3) - 48 + 21*(1350 + 66*sqrt(131)*sqrt(3))^(1/3))/((1350 + 66*sqrt(131)*sqrt(3))^(1/6)) * (((190 + 6*sqrt(393))^(2/3) + 28 + 4*(190 + 6*sqrt(393))^(1/3))/(190 + 6*sqrt(393))^(1/3)/3)^n / (n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Sep 17 2013
%t a[n_]:= a[n]= Sum[Sum[a[i]a[j-i], {i,0,j}], {j, n-3, n-1}]; a[0]=1; Table[a[n], {n, 0, 30}]
%t Flatten[{1,Table[Sum[Sum[Sum[Binomial[j,n-3*k+2*j]*Binomial[k,j] *Binomial[-m+2*k-1,k-1]/k*m,{j,0,k}],{k,m,n}],{m,1,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Sep 17 2013 *)
%o (Maxima)
%o a(n):=sum((sum(((sum(binomial(j,n-3*k+2*j)*binomial(k,j),j,0,k))* binomial(-m+2*k-1,k-1))/k,k,m,n))*m,m,1,n); /* _Vladimir Kruchinin_, May 28 2011 */
%o (Magma) I:=[1,1,3,10,35]; [n le 5 select I[n] else (3*(n-2)*Self(n-1) + (7*n-18)*Self(n-2) + 12*(n-3)*Self(n-3) + 2*(4*n-15)*Self(n-4) + 2*(2*n-9)*Self(n-5))/n: n in [1..40]]; // _G. C. Greubel_, Jun 06 2023
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A084781
%o if n==0: return 1
%o else: return sum( sum( a(k)*a(j-k) for k in range(j+1) ) for j in range(n-3,n) )
%o [a(n) for n in range(41)] # _G. C. Greubel_, Jun 06 2023
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 14 2003
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