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%I #8 Mar 30 2012 18:37:34
%S 1,1,3,21,249,4265,96773,2759229,95447863,3906844257,185703476509,
%T 10101591706057,621466309755017,42823493907501689,3278179258815893143,
%U 276843784084034075045,25636587602145118330341,2589449871609662090384097,283950074290110976428751741
%N G.f.: A(x) = x + x*ITERATE^3(x + x*ITERATE^5(x + x*ITERATE^7(x + x*ITERATE^9(x + ...)))), where ITERATE^n(F(x)) denotes the n-th iteration of F(x), and the nesting of odd iterations continue indefinitely.
%e G.f.: A(x) = x + x^2 + 3*x^3 + 21*x^4 + 249*x^5 + 4265*x^6 + 96773*x^7 +...
%e where A(x) is generated by nesting odd iterations of shifted series:
%e A(x) = x + x*B(B(B(x)));
%e B(x) = x + x*C(C(C(C(C(x)))));
%e C(x) = x + x*D(D(D(D(D(D(D(x)))))));
%e D(x) = x + x*E(E(E(E(E(E(E(E(E(x))))))))); ...
%e The coefficients in the respective series begin:
%e B: [1, 1, 5, 55, 945, 22025, 648165, 23096609, 969937489, ...];
%e C: [1, 1, 7, 105, 2373, 70553, 2580977, 111821185, 5601348627, ...];
%e D: [1, 1, 9, 171, 4797, 173913, 7637649, 391725549, 22936666197, ...];
%e E: [1, 1, 11, 253, 8481, 363209, 18648685, 1107719349, 74460444575, ...];
%e F: [1, 1, 13, 351, 13689, 676585, 39800813, 2690422761, 204504278537, ...];
%e G: [1, 1, 15, 465, 20685, 1159225, 76905145, 5833316697, 495166782107, ...];
%e H: [1, 1, 17, 595, 29733, 1863353, 137665337, 11583449749, 1086710817325, ...]; ...
%o (PARI) /* Define the n-th iteration of function F: */
%o {ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
%o /* G.f. A(x) results from nested iterations of shifted series: */
%o {a(n)=local(A=x); for(k=0, n, A=x + x*ITERATE(2*n-2*k+3, A, n)); polcoeff(A, n)}
%Y Cf. A195192, A205320.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jan 25 2012
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