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%I #9 Jan 04 2017 22:43:17
%S 1,0,-8,16,-72,640,6104,-111888,1030936,-20710880,100278648,
%T 4259777104,-94939798408,2751909451200,-58236639534824,
%U -39216365862992,24727871486377944,-1101102078005724320,44631782004871587512,-817413918683643301104,-4171285737904116490952,886660889034497668791040,-61749974510623878990175656,2477592883027309467158246512,-46339606028330299058527712104,-475992376656052398134320429920
%N E.g.f. B(x) is defined by: B(x) = 1 + Integral C(x)*A(x)^2 - C(x)^2*A(x) dx, where A(x) and C(x) are described by A278746 and A278748, respectively.
%F Given e.g.f. B(x), functions A = A(x), B = B(x), and C = C(x) satisfy:
%F (1) A^2 + B^2 + C^2 = 5,
%F (2) A^3 + B^3 + C^3 = 9.
%F (3) A^4 + B^4 + C^4 = (A'^2 + B'^2 + C'^2 + 81)/5.
%F In vector form, functions A, B, and C may be defined by
%F (4) [A,B,C] = [0,1,2] + Integral [A,B,C] X [A^2,B^2,C^2] dx.
%F Specifically,
%F (4.a) A = Integral B*C^2 - B^2*C dx,
%F (4.b) B = 1 + Integral C*A^2 - C^2*A dx,
%F (4.c) C = 2 + Integral A*B^2 - A^2*B dx.
%e E.g.f.: B(x) = 1 - 8*x^2/2! + 16*x^3/3! - 72*x^4/4! + 640*x^5/5! + 6104*x^6/6! - 111888*x^7/7! + 1030936*x^8/8! - 20710880*x^9/9! + 100278648*x^10/10! + 4259777104*x^11/11! - 94939798408*x^12/12! + 2751909451200*x^13/13! - 58236639534824*x^14/14! - 39216365862992*x^15/15! +...
%e such that B(x) = 1 + Integral C(x)*A(x)^2 - C(x)^2*A(x) dx, and A(x) and C(x) begin as follows.
%e A(x) = 2*x + 6*x^3/3! - 24*x^4/4! - 1206*x^5/5! + 8280*x^6/6! - 52858*x^7/7! + 938952*x^8/8! + 6413418*x^9/9! - 322109160*x^10/10! + 4068033894*x^11/11! - 98801788536*x^12/12! + 1090129290506*x^13/13! + 34232660705880*x^14/14! - 1166338051942842*x^15/15! +...
%e C(x) = 2 + 2*x^2/2! - 8*x^3/3! - 90*x^4/4! + 520*x^5/5! - 566*x^6/6! + 50904*x^7/7! + 396710*x^8/8! - 18390200*x^9/9! + 131511978*x^10/10! - 3005766632*x^11/11! + 33380803270*x^12/12! + 1757150970120*x^13/13! - 44857682845174*x^14/14! + 950900072932696*x^15/15! +...
%e These series satisfy:
%e (1) A(x)^2 + B(x)^2 + C(x)^2 = 5,
%e (2) A(x)^3 + B(x)^3 + C(x)^3 = 9.
%o (PARI) {a(n) = my(A=x,B=1,C=2); for(i=1,n,
%o A = intformal(B*C^2 - B^2*C +x*O(x^n));
%o B = 1 + intformal(C*A^2 - C^2*A);
%o C = 2 + intformal(A*B^2 - A^2*B);); n!*polcoeff(B,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A278746, A278748.
%K sign
%O 0,3
%A _Paul D. Hanna_, Dec 19 2016
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