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 A177409 G.f. satisfies: x = A(x) - A(x)^2 - A(A(x))^2. 1
 1, 2, 12, 108, 1208, 15536, 220832, 3390480, 55411872, 954553664, 17211258240, 323148560768, 6293245904640, 126740607526400, 2633207863038976, 56330595706808576, 1238815010325576192, 27970477203200824320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS FORMULA 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. G.f. satisfies: A(x) = A(A(x)) - A(A(x))^2 - A(A(A(x)))^2. G.f. satisfies: x = A( x-x^2 - A(x)^2 ). ... Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with: . A = 1 + xA^2 + xB^2; . B = A + xB^2 + xC^2; . C = B + xC^2 + xD^2; . D = C + xD^2 + xE^2; ... . also B = A(A(x))/x, C = A(A(A(x)))/x, D = A(A(A(A(x))))/x, etc. EXAMPLE G.f.: A(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1208*x^5 + 15536*x^6 +... Related expansions: A(x)^2 = x^2 + 4*x^3 + 28*x^4 + 264*x^5 + 2992*x^6 + 38496*x^7 +... A(A(x)) = x + 4*x^2 + 32*x^3 + 344*x^4 + 4384*x^5 + 62624*x^6 +... A(A(x))^2 = x^2 + 8*x^3 + 80*x^4 + 944*x^5 + 12544*x^6 + 182336*x^7 +... A_{-1}(x) = x - 2*x^2 - 4*x^3 - 28*x^4 - 264*x^5 - 2992*x^6 -... ... 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). COefficients in iterations A_{n}(x), n=1..8, begin: A_1: [1, 2, 12, 108, 1208, 15536, 220832, 3390480,...]; A_2: [1, 4, 32, 344, 4384, 62624, 973056, 16152608,...]; A_3: [1, 6, 60, 756, 10936, 173968, 2972320, 53760496,...]; A_4: [1, 8, 96, 1392, 22656, 399808, 7503616, 147999296,...]; A_5: [1, 10, 140, 2300, 41720, 811760, 16670112, 357673168,...]; A_6: [1, 12, 192, 3528, 70688, 1506656, 33688064, 783303776,...]; A_7: [1, 14, 252, 5124, 112504, 2610384, 63227808, 1586464432,...]; A_8: [1, 16, 320, 7136, 170496, 4281728, 111800832, 3014395008,...]. ... COefficients in squared iterations A_{n}(x)^2, for n=1..8, begin: (A_1)^2: [0, 1, 4, 28, 264, 2992, 38496, 544464, 8298080,...]; (A_2)^2: [0, 1, 8, 80, 944, 12544, 182336, 2846016, 47113792,...]; (A_3)^2: [0, 1, 12, 156, 2232, 34544, 569888, 9916112, ...]; (A_4)^2: [0, 1, 16, 256, 4320, 76800, 1429376, 27691776, ...]; (A_5)^2: [0, 1, 20, 380, 7400, 149040, 3101920, 66547024, ...]; (A_6)^2: [0, 1, 24, 528, 11664, 262912, 6064576, 143126848, ...]; (A_7)^2: [0, 1, 28, 700, 17304, 431984, 10953376, 282503760, ...]; (A_8)^2: [0, 1, 32, 896, 24512, 671744, 18586368, 520656896, ...]. PROG (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))} CROSSREFS Cf. A292810, A177397. Sequence in context: A080446 A301411 A217803 * A212273 A055897 A210997 Adjacent sequences:  A177406 A177407 A177408 * A177410 A177411 A177412 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 01 2010 STATUS approved

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Last modified May 6 16:20 EDT 2021. Contains 343586 sequences. (Running on oeis4.)