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%I #6 Mar 30 2012 18:36:57
%S 1,3,16,120,1164,13965,200960,3387636,65644780,1440018822,35314018656,
%T 958109355632,28508766348664,923461269689985,32357613376995840,
%U 1219728800410342556,49225886778689380044,2118029584754948604618
%N Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-(n+1)*x) o (1-sqrt(1-4*x))/2 } for n>=1.
%F a(n) = Sum_{j=1..n} (n+1)^(j-2)*(n-j+2)*j*(2*n-j-1)!/(n-j)!/n! - Paul D. Hanna and _Max Alekseyev_.
%e Successive self-compositions of F(x), the g.f. of A120009, begin:
%e F(x) = x + x^2 + x^3 - 6x^5 - 33x^6 - 143x^7 - 572x^8 - 2210x^9 +...
%e F(F(x)) = (1)x + 2x^2 + 4x^3 + 6x^4 - 4x^5 - 100x^6 - 664x^7 +...
%e F(F(F(x))) = x + (3)x^2 + 9x^3 + 24x^4 + 42x^5 - 87x^6 - 1575x^7 +...
%e F(F(F(F(x)))) = x + 4x^2 + (16)x^3 + 60x^4 + 192x^5 + 360x^6 +...
%e F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + (120)x^4 + 530x^5 +1955x^6 +...
%e F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 +210x^4 + (1164)x^5 +5892x^6+...
%o (PARI) {a(n)=sum(j=1, n,(n+1)^(j-2)*(n-j+2)*j*(2*n-j-1)!/(n-j)!/n!)}
%Y Cf. A120014; A120009, A127275 (g.f.=F(F(x))), A120012 (g.f.=F(F(F(x)))); A120020.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 12 2006
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