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%I #22 May 07 2024 09:15:13
%S 1,1,5,34,284,2698,28116,314558,3726504,46310523,599691087,8051875596,
%T 111674902936,1595269110991,23416737953975,352551682694096,
%U 5435771974479840,85722307390977058,1381216396128765272,22718307467700650259,381156536404199989205
%N G.f. satisfies: A(x) = x + A(A(x))^2 + A(A(x))^3.
%H Paul D. Hanna, <a href="/A211794/b211794.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) A( x - A(x)^2 - A(x)^3 ) = x.
%F (2) A( A( x - x^2 - x^3 - A(x)^2 - A(x)^3 ) ) = x. - _Paul D. Hanna_, Sep 07 2020
%F (3) A( A( A( x - x^2 - x^3 - A(x)^2 - A(x)^3 - R(x)^2 - R(x)^3 ) ) ) = x, where R(x) = x - A(x)^2 - A(x)^3 satisfies R(A(x)) = x. - _Paul D. Hanna_, May 06 2024
%e G.f.: A(x) = x + x^2 + 5*x^3 + 34*x^4 + 284*x^5 + 2698*x^6 + 28116*x^7 + ...
%e Related series begin:
%e A(A(x)) = x + 2*x^2 + 12*x^3 + 94*x^4 + 872*x^5 + 9026*x^6 + 101194*x^7 + ...
%e A(A(x))^2 = x^2 + 4*x^3 + 28*x^4 + 236*x^5 + 2264*x^6 + 23796*x^7 + ...
%e A(A(x))^3 = x^3 + 6*x^4 + 48*x^5 + 434*x^6 + 4320*x^7 + 46302*x^8 + ...
%e A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 78*x^5 + 661*x^6 + 6304*x^7 + 65624*x^8 + ...
%e A(x)^3 = x^3 + 3*x^4 + 18*x^5 + 133*x^6 + 1146*x^7 + 10995*x^8 + ...
%e where the series reversion R(x) of the g.f. A(x) begins:
%e R(x) = x - A(x)^2 - A(x)^3 = x - x^2 - 3*x^3 - 14*x^4 - 96*x^5 - 794*x^6 - 7450*x^7 - 76619*x^8 - 846161*x^9 - 9901282*x^10 + ...
%e Also, the series reversion of A(A(x)) is given by
%e x - x^2 - x^3 - A(x)^2 - A(x)^3 = x - 2*x^2 - 4*x^3 - 14*x^4 - 96*x^5 - 794*x^6 - 7450*x^7 - 76619*x^8 - 846161*x^9 - 9901282*x^10 - ...
%e Further, the series reversion of A(A(A(x))) starts as
%e x - x^2 - x^3 - A(x)^2 - A(x)^3 - R(x)^2 - R(x)^3 = x - 3*x^2 - 3*x^3 - 6*x^4 - 68*x^5 - 614*x^6 - 5952*x^7 - 62456*x^8 - 699438*x^9 - 8270469*x^10 + ...
%o (PARI) {a(n)=local(A=x+x^2, B=x); for(i=1, n, B=subst(A, x, A+x*O(x^n)); A=x+B^2+B^3); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2-A^3+x*O(x^n))); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A190761, A171991, A213591.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jun 27 2012
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