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A177397 G.f. satisfies: x = A(x) - A(A(x))^2 - A(A(A(x)))^2. 1

%I #2 Mar 30 2012 18:37:21

%S 1,2,20,316,6312,146256,3765792,105104272,3130299744,98434722240,

%T 3243746014592,111400312737152,3970597596057856,146403897677390336,

%U 5570169496704513024,218228733514994839808,8789314898568643716608

%N G.f. satisfies: x = A(x) - A(A(x))^2 - A(A(A(x)))^2.

%F G.f. A(x) satisfies: A_{n}(x) = A_{n+1}(x) - A_{n+2}(x)^2 - A_{n+3}(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(A(x)))^2 - A(A(A(A(x))))^2.

%F G.f. satisfies: x = A( x - A(x)^2 - A(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 + xB^2 + xC^2;

%F . B = A + xC^2 + xD^2;

%F . C = B + xD^2 + xE^2;

%F . D = C + xE^2 + xF^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 + 20*x^3 + 316*x^4 + 6312*x^5 + 146256*x^6 +...

%e Related expansions:

%e A(A(x)) = x + 4*x^2 + 48*x^3 + 840*x^4 + 18016*x^5 + 440992*x^6 +...

%e A(A(A(x))) = x + 6*x^2 + 84*x^3 + 1620*x^4 + 37352*x^5 +969328*x^6 +...

%e A_{-1}(x) = x - 2*x^2 - 12*x^3 - 156*x^4 - 2776*x^5 - 59344*x^6 -...

%e A_{-2}(x) = x - 4*x^2 - 16*x^3 - 200*x^4 - 3488*x^5 - 73632*x^6 -...

%e ...

%e Illustrate A_{n}(x) = A_{n+1}(x) - A_{n+2}(x)^2 - A_{n+3}(x)^2 by the following tables of coefficients in the iterations of g.f. A(x).

%e Coefficients in iterations A_{n}(x), for n=1..8, begin:

%e A_1: [1, 2, 20, 316, 6312, 146256, 3765792, 105104272,...];

%e A_2: [1, 4, 48, 840, 18016, 440992, 11875712, 344335328,...];

%e A_3: [1, 6, 84, 1620, 37352, 969328, 27429152, 830501936,...];

%e A_4: [1, 8, 128, 2704, 66944, 1843776, 54945792, 1742374336,...];

%e A_5: [1, 10, 180, 4140, 109800, 3208080, 100748064, 3350443472,...];

%e A_6: [1, 12, 240, 5976, 169312, 5241056, 173389696, 6048725920,...];

%e A_7: [1, 14, 308, 8260, 249256, 8160432, 284130336, 10393259632,...];

%e A_8: [1, 16, 384, 11040, 353792, 12226688, 447456256, 17147935616,...].

%e ...

%e Coefficients in squared iterations A_{n}(x)^2, for n=1..8, begin:

%e (A_1)^2: [0, 1, 4, 44, 712, 14288, 330400, 8468944, 235111136,...];

%e (A_2)^2: [0, 1, 8, 112, 2064, 45056, 1106752, 29714496, 856278464,...];

%e (A_3)^2: [0, 1, 12, 204, 4248, 101200, 2659040, 75389776, ...];

%e (A_4)^2: [0, 1, 16, 320, 7456, 193536, 5450880, 163841280, ...];

%e (A_5)^2: [0, 1, 20, 460, 11880, 334800, 10102560, 322325328, ...];

%e (A_6)^2: [0, 1, 24, 624, 17712, 539648, 17414080, 589547072, ...];

%e (A_7)^2: [0, 1, 28, 812, 25144, 824656, 28388192, 1018522064, ...];

%e (A_8)^2: [0, 1, 32, 1024, 34368, 1208320, 44253440, 1679760384, ...].

%o (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x-(A+x*O(x^n))^2-subst(A,x,A+x*O(x^n))^2)); polcoeff(A, n))}

%Y Cf. A177395, A177396.

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 31 2010

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