%I #24 Sep 09 2017 09:48:19

%S 6,11,48,262,2148,20504,246972,3362728,53193732,937974536,18476814108,

%T 399141493432,9422633062788,240736366703624,6627438693162012,

%U 195420759324590008,6147604272502028292,205458180805221547016,7270950279223014832668,271596882085548398880952,10679317246361342569073028,440906517489248538430081544,19070223396309931435004013852,862320222118766033966860149688

%N E.g.f.: A(x) + B(x) + C(x) where A'(x) = B(x)*C(x), B'(x) = A(x)*C(x), and C'(x) = A(x)*B(x), where A(x), B(x), and C(x) are the e.g.f.s of A292121, A292122, and A292123, respectively.

%H Paul D. Hanna, <a href="/A292120/b292120.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. A(x) + B(x) + C(x) and related functions A(x), B(x), and C(x) satisfy:

%F (1a) A(x) + B(x) + C(x) = d/dx log( ((A(x) + B(x))*(A(x) + C(x))*(B(x) + C(x)))/60 ).

%F (1b) A(x) + B(x) + C(x) = (5*exp(Integral A(x) dx) + 4*exp(Integral B(x) dx) + 3*exp(Integral C(x) dx))/2.

%F (2a) A(x) = 1 + Integral B(x)*C(x) dx.

%F (2b) B(x) = 2 + Integral A(x)*C(x) dx.

%F (2c) C(x) = 3 + Integral A(x)*B(x) dx.

%F (3a) B(x)^2 - A(x)^2 = 3.

%F (3b) C(x)^2 - A(x)^2 = 8.

%F (3c) C(x)^2 - B(x)^2 = 5.

%F (4a) A(x) = (B'(x) + C'(x))/(B(x) + C(x)).

%F (4b) B(x) = (C'(x) + A'(x))/(C(x) + A(x)).

%F (4c) C(x) = (A'(x) + B'(x))/(A(x) + B(x)).

%F (5a) A(x) + B(x) = 3 * exp( Integral C(x) dx ).

%F (5b) A(x) + C(x) = 4 * exp( Integral B(x) dx ).

%F (5c) B(x) + C(x) = 5 * exp( Integral A(x) dx ).

%e E.g.f.: A(x) + B(x) + C(x) = 6 + 11*x + 48*x^2/2! + 262*x^3/3! + 2148*x^4/4! + 20504*x^5/5! + 246972*x^6/6! + 3362728*x^7/7! + 53193732*x^8/8! + 937974536*x^9/9! + 18476814108*x^10/10! +...

%e Related series.

%e 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 where A(x) = 1 + Integral B(x)*C(x) dx.

%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 C(x) = (A'x) + B'(x)) / (A(x) + B(x)).

%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 B(x) = (A'x) + C'(x)) / (A(x) + C(x)).

%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 A(x) = (B'x) + C'(x)) / (B(x) + C(x)).

%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+B+C,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A292121 (A), A292122 (B), A292123 (C), A292124 (A*B*C).

%K nonn

%O 0,1

%A _Paul D. Hanna_, Sep 08 2017