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%I #26 Sep 09 2017 09:46:57
%S 1,6,13,102,653,7134,80257,1138638,17577977,314204406,6141247573,
%T 133263548022,3137974308293,80288176882254,2208474466924297,
%U 65151554971629918,2048997857627015537,68489950399363334886,2423571453722122287133,90533973988868134321542,3559734642493103582865533,146969730550281362048202174,6356719089202681283092805137,287440643937159436055153903598,13562631117348347165659535163497
%N E.g.f. A(x) satisfies: A'(x) = B(x)*C(x) such that B(x)^2 - A(x)^2 = 3 and C(x)^2 - A(x)^2 = 8, where B(x) and C(x) are the e.g.f.s of A292122 and A292123, respectively.
%H Paul D. Hanna, <a href="/A292121/b292121.txt">Table of n, a(n) for n = 0..300</a>
%F E.g.f. A(x) and related functions B(x) and C(x) satisfy:
%F (1a) A(x) = 1 + Integral B(x)*C(x) dx.
%F (1b) B(x) = 2 + Integral A(x)*C(x) dx.
%F (1c) C(x) = 3 + Integral A(x)*B(x) dx.
%F (2a) B(x)^2 - A(x)^2 = 3.
%F (2b) C(x)^2 - A(x)^2 = 8.
%F (2c) C(x)^2 - B(x)^2 = 5.
%F (3a) A(x) = (B'(x) + C'(x))/(B(x) + C(x)).
%F (3b) B(x) = (C'(x) + A'(x))/(C(x) + A(x)).
%F (3c) C(x) = (A'(x) + B'(x))/(A(x) + B(x)).
%F (4a) A(x) + B(x) = 3 * exp( Integral C(x) dx ).
%F (4b) A(x) + C(x) = 4 * exp( Integral B(x) dx ).
%F (4c) B(x) + C(x) = 5 * exp( Integral A(x) dx ).
%F (5a) A(x) = (-5*exp(Integral A(x) dx) + 4*exp(Integral B(x) dx) + 3*exp(Integral C(x) dx))/2.
%F (5b) B(x) = (5*exp(Integral A(x) dx) - 4*exp(Integral B(x) dx) + 3*exp(Integral C(x) dx))/2.
%F (5c) C(x) = (5*exp(Integral A(x) dx) + 4*exp(Integral B(x) dx) - 3*exp(Integral C(x) dx))/2.
%F (6a) A(x)^m = 1 + Integral m * A(x)^(m-1) * B(x) * C(x) dx.
%F (6b) B(x)^m = 2^m + Integral m * A(x) * B(x)^(m-1) * C(x) dx.
%F (6c) C(x)^m = 3^m + Integral m * A(x) * B(x) * C(x)^(m-1) dx.
%e E.g.f. A(x) = 1 + 6*x + 13*x^2/2! + 102*x^3/3! + 653*x^4/4! + 7134*x^5/5! + 80257*x^6/6! + 1138638*x^7/7! + 17577977*x^8/8! + 314204406*x^9/9! + 6141247573*x^10/10! +...
%e Related series.
%e B(x) = 2 + 3*x + 20*x^2/2! + 78*x^3/3! + 736*x^4/4! + 6672*x^5/5! + 83360*x^6/6! + 1113072*x^7/7! + 17810944*x^8/8! + 311847168*x^9/9! + 6167567360*x^10/10! +...
%e where B(x)^2 - A(x)^2 = 3.
%e C(x) = 3 + 2*x + 15*x^2/2! + 82*x^3/3! + 759*x^4/4! + 6698*x^5/5! + 83355*x^6/6! + 1111018*x^7/7! + 17804811*x^8/8! + 311922962*x^9/9! + 6167999175*x^10/10! +...
%e where C(x)^2 - A(x)^2 = 8.
%e A(x) + B(x) = 3 + 9*x + 33*x^2/2! + 180*x^3/3! + 1389*x^4/4! + 13806*x^5/5! + 163617*x^6/6! + 2251710*x^7/7! + 35388921*x^8/8! + 626051574*x^9/9! + 12308814933*x^10/10! +...
%e where A(x) + B(x) = (A'(x) + B'(x)) / C(x),
%e and A(x) + B(x) = 3 * exp( Integral C(x) dx ).
%e A(x) + C(x) = 4 + 8*x + 28*x^2/2! + 184*x^3/3! + 1412*x^4/4! + 13832*x^5/5! + 163612*x^6/6! + 2249656*x^7/7! + 35382788*x^8/8! + 626127368*x^9/9! + 12309246748*x^10/10! +...
%e where A(x) + C(x) = (A'(x) + C'(x)) / B(x),
%e and A(x) + C(x) = 4 * exp( Integral B(x) dx ).
%e B(x) + C(x) = 5 + 5*x + 35*x^2/2! + 160*x^3/3! + 1495*x^4/4! + 13370*x^5/5! + 166715*x^6/6! + 2224090*x^7/7! + 35615755*x^8/8! + 623770130*x^9/9! + 12335566535*x^10/10! +...
%e where B(x) + C(x) = (B'(x) + C'(x)) / A(x),
%e and B(x) + C(x) = 5 * exp( Integral A(x) dx ).
%o (PARI) {a(n) = my(A=1,B=2,C=3); for(i=0,n, A = 1 + intformal(B*C +x*O(x^n)); B = 2 + intformal(A*C); C = 3 + intformal(A*B)); n!*polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A292120 (A+B+C), A292122 (B), A292123 (C), A292124 (A*B*C).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 08 2017
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