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%I #7 May 21 2014 13:24:35
%S 1,2,9,68,710,9348,148085,2740672,58033953,1383923040,36705564368,
%T 1071911496576,34179790156473,1181725089179936,44035415728886145,
%U 1759481180119564288,75042973200676887772,3402984761691650083008
%N Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) } for n>=1.
%C Main diagonal of A120019, the table of self-compositions of A120010.
%C For n>=1, n divides a(n): a(n)/n = A120022(n).
%F a(n) = Sum_{j=1..n} Catalan(n-j) * [ Sum_{i=1..j} (-1)^(j-i) * n^(i-1) * C(n-j+i, j-i) * C(n-j+i-1, i-1) ];
%F a(n) = Sum_{j=0..n-1} n^j * [ Sum_{i=j..n-1} (-1)^(i-j) * Catalan(n-i-1) * C(n-i+j, i-j) * C(n-i+j-1, j) ], where Catalan(n) = A000108(n) = C(2n, n)/(n+1).
%e Successive iterations of F(x), the g.f. of A120010, begin:
%e F(x) = (1)x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
%e F(F(x)) = x + (2)x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
%e F(F(F(x))) = x + 3x^2 + (9)x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
%e F(F(F(F(x)))) = x + 4x^2 + 16x^3 + (68)x^4 + 312x^5 + 1536x^6 +...
%e F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 130x^4 + (710)x^5 + 4070x^6 +...
%e F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + 1416x^5 + (9348)x^6+..
%o (PARI) {a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)*n^(i-1)))}
%o for(n=1,25,print1(a(n),", "))
%Y Cf. A120010, A120019, A120021, A120022.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 14 2006
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