_Paul D. Hanna (pauldhanna(AT)juno.com), _, Sep 11 2011

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G.f.: A(x) = 1 x + x ^2 + 2*x^2 3 + 8*x^3 4 + 52*x^4 5 + 480*x^5 6 + 5811*x^6 7 +...

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G.f.: A(x) = x + x*ITERATE^2(x + x*ITERATE^3(x + x*ITERATE^4(x + x*ITERATE^5(x + ...)))), where ITERATE^n(F(x)) denotes the n-th iteration of F(x), and the nested iterations continue indefinitely.

Paul D. Hanna, <a href="/A195192/b195192.txt">Table of n, a(n) for n = 1..100</a>

G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 52*x^4 + 480*x^5 + 5811*x^6 +...

(PARI) /* Define the n-th iteration of function F: */

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allocated for Paul D. Hanna

G.f.: A(x) = x + x*ITERATE^2(x + x*ITERATE^3(x + x*ITERATE^4(x + x*ITERATE^5(x + ...)))), where ITERATE^n(F(x)) denotes the n-th iteration of F(x), and the nested iterations continue indefinitely.

1, 1, 2, 8, 52, 480, 5811, 87768, 1599189, 34317476, 851848787, 24117965057, 770085965621, 27472955220394, 1086491917437511, 47313129973804599, 2255470786017010440, 117103678748934306726, 6591945250191262271448, 400690880212910627734513, 26204671530683391316131317

1,3

G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 52*x^4 + 480*x^5 + 5811*x^6 +...

where A(x) is generated by nested iterations of shifted series:

A(x) = x + x*B(B(x)), where

B(x) = x + x^2 + 3*x^3 + 18*x^4 + 165*x^5 + 2034*x^6 + 31500*x^7 +...;

B(x) = x + x*C(C(C(x))), where

C(x) = x + x^2 + 4*x^3 + 32*x^4 + 380*x^5 + 5904*x^6 + 112528*x^7 +...;

C(x) = x + x*D(D(D(D(x)))), where

D(x) = x + x^2 + 5*x^3 + 50*x^4 + 730*x^5 + 13720*x^6 + 311730*x^7 +...;

D(x) = x + x*E(E(E(E(E(x))))), where

E(x) = x + x^2 + 6*x^3 + 72*x^4 + 1248*x^5 + 27552*x^6 + 728175*x^7 +...;

E(x) = x + x*F(F(F(F(F(F(x)))))), where

F(x) = x + x^2 + 7*x^3 + 98*x^4 + 1967*x^5 + 49910*x^6 + 1505546*x^7 +...;

F(x) = x + x*G(G(G(G(G(G(G(x))))))), where

G(x) = x + x^2 + 8*x^3 + 128*x^4 + 2920*x^5 + 83744*x^6 + 2840520*x^7 +...;

G(x) = x + x*H(H(H(H(H(H(H(H(x)))))))), where

H(x) = x + x^2 + 9*x^3 + 162*x^4 + 4140*x^5 + 132444*x^6 + 4991148*x^7 +...; ...

(PARI) /* Define the n-th iteration of function F: */

{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}

/* A(x) results from nested iterations of shifted series: */

{a(n)=local(A=x); for(k=0, n, A=ITERATE(n-k+1, x + x*A, n)); polcoeff(A, n)}

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Paul D. Hanna (pauldhanna(AT)juno.com), Sep 11 2011

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