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%I #10 Aug 24 2017 17:59:59
%S 1,2,9,60,530,5892,79681,1276760,23729310,502780580,11974950746,
%T 316917570312,9230453871756,293492484431720,10117826259791025,
%U 375952605020796720,14980065429077943734,637215061582781559972
%N Coefficients of x^n in the n-th iteration of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2 } for n>=1.
%C a(n) is divisible by n for n>=1; a(n)/n = A120016(n).
%C Main diagonal of table (A120013) of iterations of A120009.
%F a(n) = [x^n] x*((1-n+n^2) - n^2*(n+1)*x - n*(1-(n+2)*x)*C(x) )/(1-n+n^2*x)^2, where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).
%F a(n) = n^(n-1) - Sum_{k=2..n-2} n^(k-1)*k*(k-1)*(n-k-1)*(2*n-k-2)!/(n-k)!/n!
%e Successive iterations of F(x), the g.f. of A120009, begin:
%e F(x) = (1)x + x^2 + x^3 - 6x^5 - 33x^6 - 143x^7 - 572x^8 - 2210x^9 +...
%e F(F(x)) = x + (2)x^2 + 4x^3 + 6x^4 - 4x^5 - 100x^6 - 664x^7 +...
%e F(F(F(x))) = x + 3x^2 + (9)x^3 + 24x^4 + 42x^5 - 87x^6 - 1575x^7 +...
%e F(F(F(F(x)))) = x + 4x^2 + 16x^3 + (60)x^4 + 192x^5 + 360x^6 +...
%e F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 120x^4 + (530)x^5 +1955x^6 +...
%e F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 +210x^4 +1164x^5 + (5892)x^6 +...
%o (PARI) a(n)=local(k=n,x=X+X^3*O(X^n));polcoeff( x*((1-k+k^2)-k^2*(k+1)*x-k*(1-(k+2)*x)*(1-sqrt(1-4*x))/2/x)/(1-k+k^2*x)^2,n,X)
%o (PARI) /* Generated as the n-th self-composition of A120009: */ a(n)=local(F=((1-3*x)*sqrt(1-4*x+x^3*O(x^n)) - (1-x)*(1-4*x))/(2*x^2), G=x+x*O(x^n)); if(n<1, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, n, x)))
%o (PARI) a(n)=n^(n-1)-sum(k=2,n-2,n^(k-1)*k*(k-1)*(n-k-1)*(2*n-k-2)!/(n-k)!)/n!
%Y Cf. A120016 (a(n)/n); A120009, A127275 (g.f.=F(F(x))), A120012 (g.f.=F(F(F(x)))); A000108 (Catalan); A120015, A120020, A120013.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 07 2006, Jun 09 2006
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