A292120 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.

6, 11, 48, 262, 2148, 20504, 246972, 3362728, 53193732, 937974536, 18476814108, 399141493432, 9422633062788, 240736366703624, 6627438693162012, 195420759324590008, 6147604272502028292, 205458180805221547016, 7270950279223014832668, 271596882085548398880952, 10679317246361342569073028, 440906517489248538430081544, 19070223396309931435004013852, 862320222118766033966860149688

Offset

0,1

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

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

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

(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.

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

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

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

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

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

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

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

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

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

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

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

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

Example

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! +...
Related series.
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! +...
where A(x) = 1 + Integral B(x)*C(x) dx.
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! +...
where B(x)^2 - A(x)^2 = 3.
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! +...
where C(x)^2 - A(x)^2 = 8.
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! +...
where C(x) =  (A'x) + B'(x)) / (A(x) + B(x)).
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! +...
where B(x) =  (A'x) + C'(x)) / (A(x) + C(x)).
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! +...
where A(x) =  (B'x) + C'(x)) / (B(x) + C(x)).

Prog

(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)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

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

Sequence in context: A269816 A270020 A269878 * A099437 A368373 A271093

Adjacent sequences: A292117 A292118 A292119 * A292121 A292122 A292123

Keyword

nonn

Author

Paul D. Hanna, Sep 08 2017

Status

approved