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%I #8 Nov 29 2013 12:32:43
%S 1,-1,3,-12,55,-273,1431,-7837,44726,-266381,1658300,-10768609,
%T 72407500,-498510748,3477131466,-24461950817,174793982029,
%U -1294469021982,10177610535232,-85391010070808,741460526149745,-6291905077685633
%N G.f. F_1(x) satisfies: x = Sum_{n>=1} F_{n}(x)^n, where the n-th iteration of the g.f. is defined by: F_{n}(x) = F_{n-1}( F_1(x) ) with F_0(x) = x.
%H Paul D. Hanna, <a href="/A171780/b171780.txt">Table of n, a(n), n = 1..100.</a>
%e G.f.: F_1(x) = x - x^2 + 3*x^3 - 12*x^4 + 55*x^5 - 273*x^6 + 1431*x^7 +...
%e Let F_{n}(x) denote the n-th iteration of g.f. F_1(x), then:
%e (1) x = F_1(x)^1 + F_2(x)^2 + F_3(x)^3 + F_4(x)^4 + F_5(x)^5 +...
%e (2) F_n(x) = F_{n+1}(x) + F_{n+2}(x)^2 + F_{n+3}(x)^3 + F_{n+4}(x)^4 +...
%e The initial iterations of the g.f. begin:
%e F_2(x) = x - 2*x^2 + 8*x^3 - 40*x^4 + 224*x^5 - 1345*x^6 +...
%e F_3(x) = x - 3*x^2 + 15*x^3 - 90*x^4 + 595*x^5 - 4184*x^6 +...
%e F_4(x) = x - 4*x^2 + 24*x^3 - 168*x^4 + 1280*x^5 - 10306*x^6 +...
%e F_5(x) = x - 5*x^2 + 35*x^3 - 280*x^4 + 2415*x^5 - 21895*x^6 +...
%e F_6(x) = x - 6*x^2 + 48*x^3 - 432*x^4 + 4160*x^5 - 41923*x^6 +...
%e F_7(x) = x - 7*x^2 + 63*x^3 - 630*x^4 + 6699*x^5 - 74270*x^6 +...
%e F_8(x) = x - 8*x^2 + 80*x^3 - 880*x^4 + 10240*x^5 - 123844*x^6 +...
%e where initial terms of F_n(x)^n for n=1..8 begin:
%e F_1(x)^1 = x - x^2 + 3*x^3 - 12*x^4 + 55*x^5 - 273*x^6 + 1431*x^7 +...
%e F_2(x)^2 = x^2 - 4*x^3 + 20*x^4 - 112*x^5 + 672*x^6 - 4226*x^7 +...
%e F_3(x)^3 = x^3 - 9*x^4 + 72*x^5 - 567*x^6 + 4485*x^7 - 35817*x^8 +...
%e F_4(x)^4 = x^4 - 16*x^5 + 192*x^6 - 2080*x^7 + 21504*x^8 +...
%e F_5(x)^5 = x^5 - 25*x^6 + 425*x^7 - 6150*x^8 + 81700*x^9 +...
%e F_6(x)^6 = x^6 - 36*x^7 + 828*x^8 - 15552*x^9 + 260400*x^10 +...
%e F_7(x)^7 = x^7 - 49*x^8 + 1470*x^9 - 34937*x^10 + 723632*x^11 +...
%e F_8(x)^8 = x^8 - 64*x^9 + 2432*x^10 - 71552*x^11 + 1802240*x^12 +...
%o (PARI) {a(n)=local(a_n=0,G=x,F=x-x^2+sum(k=3,n-1,a(k)*x^k));if(n<1,0,if(n==1,1, for(k=2,n,G=x;for(i=1,k,G=subst(F,x,G+x*O(x^n)));a_n=a_n-polcoeff(G^k,n));a_n))}
%K sign
%O 1,3
%A _Paul D. Hanna_, Jan 24 2010
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