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A278747
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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.
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5
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1, 0, -8, 16, -72, 640, 6104, -111888, 1030936, -20710880, 100278648, 4259777104, -94939798408, 2751909451200, -58236639534824, -39216365862992, 24727871486377944, -1101102078005724320, 44631782004871587512, -817413918683643301104, -4171285737904116490952, 886660889034497668791040, -61749974510623878990175656, 2477592883027309467158246512, -46339606028330299058527712104, -475992376656052398134320429920
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OFFSET
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0,3
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LINKS
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FORMULA
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Given e.g.f. B(x), functions A = A(x), B = B(x), and C = C(x) satisfy:
(1) A^2 + B^2 + C^2 = 5,
(2) A^3 + B^3 + C^3 = 9.
(3) A^4 + B^4 + C^4 = (A'^2 + B'^2 + C'^2 + 81)/5.
In vector form, functions A, B, and C may be defined by
(4) [A,B,C] = [0,1,2] + Integral [A,B,C] X [A^2,B^2,C^2] dx.
Specifically,
(4.a) A = Integral B*C^2 - B^2*C dx,
(4.b) B = 1 + Integral C*A^2 - C^2*A dx,
(4.c) C = 2 + Integral A*B^2 - A^2*B dx.
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EXAMPLE
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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! +...
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.
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! +...
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! +...
These series satisfy:
(1) A(x)^2 + B(x)^2 + C(x)^2 = 5,
(2) A(x)^3 + B(x)^3 + C(x)^3 = 9.
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PROG
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(PARI) {a(n) = my(A=x, B=1, C=2); for(i=1, n,
A = intformal(B*C^2 - B^2*C +x*O(x^n));
B = 1 + intformal(C*A^2 - C^2*A);
C = 2 + intformal(A*B^2 - A^2*B); ); n!*polcoeff(B, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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