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G.f. A(x) satisfies: A(x) = x + x*A(x) + x^2*A(A(x)) + x^3*A(A(A(x))) +...
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%I #5 Nov 21 2022 00:37:00

%S 1,1,2,5,15,53,215,976,4859,26150,150585,920910,5946929,40369352,

%T 287020631,2130932767,16478548793,132438164617,1104141400679,

%U 9532801486793,85102769453094,784511536839904,7458380835336557

%N G.f. A(x) satisfies: A(x) = x + x*A(x) + x^2*A(A(x)) + x^3*A(A(A(x))) +...

%C Equals row sums and column 1 of triangle A120568 of self-compositions of A(x).

%H Paul D. Hanna, <a href="/A120567/b120567.txt">Table of n, a(n) for n = 1..200</a>

%e The successive self-compositions of the g.f. A(x) begin:

%e A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 53*x^6 + 215*x^7 + 976*x^8+...

%e A(A(x)) = x + 2*x^2 + 6*x^3 + 21*x^4 + 82*x^5 + 351*x^6 + 1630*x^7 +...

%e A(A(A(x))) = x + 3*x^2 + 12*x^3 + 54*x^4 + 263*x^5 + 1364*x^6 +...

%e A(A(A(A(x)))) = x + 4*x^2 + 20*x^3 + 110*x^4 + 644*x^5 + 3956*x^6 +...

%e A(A(A(A(A(x))))) = x + 5*x^2 + 30*x^3 + 195*x^4 + 1335*x^5 +9505*x^6+...

%e These g.f.s form the columns of triangle A120568.

%o (PARI) {a(n)=local(F=x+x*O(x^n), G=F, H=x); for(m=1, n, for(k=1, m, G=subst(F, x, G); H=H+x^k*truncate(G) +x*O(x^n)); F=H; G=x+x^2*O(x^m); H=G;);polcoeff(F, n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A120568.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jun 14 2006