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%I #6 Nov 18 2022 20:07:05
%S 1,2,12,108,1208,15536,220832,3390480,55411872,954553664,17211258240,
%T 323148560768,6293245904640,126740607526400,2633207863038976,
%U 56330595706808576,1238815010325576192,27970477203200824320
%N G.f. satisfies: x = A(x) - A(x)^2 - A(A(x))^2.
%F G.f. satisfies: A_{n}(x) = A_{n+1}(x) - A_{n+1}(x)^2 - A_{n+2}(x)^2 where A_{n+1}(x) = A_{n}(A(x)) denotes iteration with A_0(x)=x.
%F G.f. satisfies: A(x) = A(A(x)) - A(A(x))^2 - A(A(A(x)))^2.
%F G.f. satisfies: x = A( x-x^2 - A(x)^2 ).
%F ...
%F Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
%F . A = 1 + xA^2 + xB^2;
%F . B = A + xB^2 + xC^2;
%F . C = B + xC^2 + xD^2;
%F . D = C + xD^2 + xE^2; ...
%F . also B = A(A(x))/x, C = A(A(A(x)))/x, D = A(A(A(A(x))))/x, etc.
%e G.f.: A(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1208*x^5 + 15536*x^6 +...
%e Related expansions:
%e A(x)^2 = x^2 + 4*x^3 + 28*x^4 + 264*x^5 + 2992*x^6 + 38496*x^7 +...
%e A(A(x)) = x + 4*x^2 + 32*x^3 + 344*x^4 + 4384*x^5 + 62624*x^6 +...
%e A(A(x))^2 = x^2 + 8*x^3 + 80*x^4 + 944*x^5 + 12544*x^6 + 182336*x^7 +...
%e A_{-1}(x) = x - 2*x^2 - 4*x^3 - 28*x^4 - 264*x^5 - 2992*x^6 -...
%e ...
%e Illustrate A_{n}(x) = A_{n+1}(x) - A_{n+1}(x)^2 - A_{n+2}(x)^2 by the following tables of coefficients in the iterations of g.f. A(x).
%e COefficients in iterations A_{n}(x), n=1..8, begin:
%e A_1: [1, 2, 12, 108, 1208, 15536, 220832, 3390480,...];
%e A_2: [1, 4, 32, 344, 4384, 62624, 973056, 16152608,...];
%e A_3: [1, 6, 60, 756, 10936, 173968, 2972320, 53760496,...];
%e A_4: [1, 8, 96, 1392, 22656, 399808, 7503616, 147999296,...];
%e A_5: [1, 10, 140, 2300, 41720, 811760, 16670112, 357673168,...];
%e A_6: [1, 12, 192, 3528, 70688, 1506656, 33688064, 783303776,...];
%e A_7: [1, 14, 252, 5124, 112504, 2610384, 63227808, 1586464432,...];
%e A_8: [1, 16, 320, 7136, 170496, 4281728, 111800832, 3014395008,...].
%e ...
%e COefficients in squared iterations A_{n}(x)^2, for n=1..8, begin:
%e (A_1)^2: [0, 1, 4, 28, 264, 2992, 38496, 544464, 8298080,...];
%e (A_2)^2: [0, 1, 8, 80, 944, 12544, 182336, 2846016, 47113792,...];
%e (A_3)^2: [0, 1, 12, 156, 2232, 34544, 569888, 9916112, ...];
%e (A_4)^2: [0, 1, 16, 256, 4320, 76800, 1429376, 27691776, ...];
%e (A_5)^2: [0, 1, 20, 380, 7400, 149040, 3101920, 66547024, ...];
%e (A_6)^2: [0, 1, 24, 528, 11664, 262912, 6064576, 143126848, ...];
%e (A_7)^2: [0, 1, 28, 700, 17304, 431984, 10953376, 282503760, ...];
%e (A_8)^2: [0, 1, 32, 896, 24512, 671744, 18586368, 520656896, ...].
%o (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x-x^2-(A+x*O(x^n))^2)); polcoeff(A, n))}
%Y Cf. A292810, A177397, A276365.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 01 2010
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