A292122 E.g.f. B(x) satisfies: B'(x) = A(x)*C(x) such that B(x)^2 - A(x)^2 = 3 and C(x)^2 - B(x)^2 = 5, where A(x) and C(x) are the e.g.f.s of A292121 and A292123, respectively.

2, 3, 20, 78, 736, 6672, 83360, 1113072, 17810944, 311847168, 6167567360, 132939299328, 3142344933376, 80224248225792, 2209481228472320, 65134576849385472, 2049303151562850304, 68484117776544104448, 2423689450690189721600, 90531454034353494294528, 3559791292388307701334016, 146968393370524340544602112, 6356752154870738583164026880, 287439789131448083136535068672, 13562654176394604163638350577664

Offset

0,1

Links

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

Formula

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

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

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

(1c) C(x) = 3 + Integral A(x)*B(x) dx.

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

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

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

(3a) A(x) = (B'(x) + C'(x))/(B(x) + C(x)).

(3b) B(x) = (C'(x) + A'(x))/(C(x) + A(x)).

(3c) C(x) = (A'(x) + B'(x))/(A(x) + B(x)).

(4a) A(x) + B(x) = 3 * exp( Integral C(x) dx ).

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

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

(5a) A(x) = (-5*exp(Integral A(x) dx) + 4*exp(Integral B(x) dx) + 3*exp(Integral C(x) dx))/2.

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

(5c) C(x) = (5*exp(Integral A(x) dx) + 4*exp(Integral B(x) dx) - 3*exp(Integral C(x) dx))/2.

(6a) A(x)^m = 1 + Integral m * A(x)^(m-1) * B(x) * C(x) dx.

(6b) B(x)^m = 2^m + Integral m * A(x) * B(x)^(m-1) * C(x) dx.

(6c) C(x)^m = 3^m + Integral m * A(x) * B(x) * C(x)^(m-1) dx.

Example

E.g.f. 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! +...
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 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 - B(x)^2 = 5.
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 A(x) + B(x) =  (A'(x) + B'(x)) / C(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 A(x) + C(x) =  (A'(x) + C'(x)) / B(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 B(x) + C(x) =  (B'(x) + C'(x)) / A(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(B, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A292120 (A+B+C), A292121 (A), A292123 (C), A292124 (A*B*C).

Sequence in context: A041567 A338084 A087301 * A264417 A348311 A066166

Adjacent sequences: A292119 A292120 A292121 * A292123 A292124 A292125

Keyword

nonn

Author

Paul D. Hanna, Sep 08 2017

Status

approved