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%I #18 Nov 20 2017 10:50:05
%S 0,1,0,1,3,5,22,64,198,710,2332,8105,28665,100653,361104,1301180,
%T 4713267,17217021,63140534,232702261,861507251,3200666821,11933894310,
%U 44636509320,167427781950,629691033738,2373987233286,8970240131032,33965443165016,128857452216256
%N G.f. A(x) satisfies A(x)+A(x)^2+A(x)^3 = (1-sqrt(1-4*x))/2.
%H Alois P. Heinz, <a href="/A212260/b212260.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = (sum(k=2..n, C(2*n-k-1,n-1)*(sum(i=1..k-1, (-1)^i*C(i,k-i-1) * C(k+i-1,k-1)))) +C(2*n-2,n-1))/n if n>0, a(0) = 0.
%F a(n) ~ c * 4^n / (sqrt(Pi) * n^(3/2)), where c = 0.12273243737616788383659461976... is the real root of the equation 8*c*(134*c^2 - 1) = 1. - _Vaclav Kotesovec_, Nov 20 2017
%p a:= n-> coeff(series(RootOf(A+A^2+A^3=(1-sqrt(1-4*x))/2, A), x, n+1), x, n): seq(a(n), n=0..40); # _Alois P. Heinz_, May 12 2012
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n<5, [0, 1, 0, 1, 3][n+1],
%p (2*(n-1)*(n-2)*(n-3)*(21574*n^2-148237*n+252420)*a(n-1)
%p +(n-2)*(n-3)*(30485*n^3-173514*n^2+191353*n+116820)*a(n-2)
%p -4*(n-3)*(12730*n^4-266121*n^3+1766621*n^2-4771248*n+4563630)*a(n-3)
%p -36*(6*n-23)*(67*n-183)*(6*n-25)*(3*n-10)*(3*n-11)*a(n-4))/
%p (132*(67*n-250)*(n-1)*(n-2)*(n-3)*n))
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 27 2013
%t nn=29;f[x_]:=Sum[a[n]x^n,{n,0,nn}];sol=SolveAlways[0==Series[f[x]+f[x]^2 +f[x]^3-(1-(1-4x)^(1/2))/2,{x,0,nn}],x][[3]];Table[a[n],{n,0,nn}]/.sol (* _Geoffrey Critzer_, Sep 27 2013 *)
%o (Maxima) a(n):=(sum(binomial(2*n-k-1,n-1)*(sum((-1)^i*binomial(i,k-i-1) *binomial(k+i-1,k-1), i,1,k-1)), k,2,n) +binomial(2*n-2,n-1))/n;
%Y Cf. A103779.
%K nonn
%O 0,5
%A _Vladimir Kruchinin_, May 12 2012
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