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A292121 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. 5
1, 6, 13, 102, 653, 7134, 80257, 1138638, 17577977, 314204406, 6141247573, 133263548022, 3137974308293, 80288176882254, 2208474466924297, 65151554971629918, 2048997857627015537, 68489950399363334886, 2423571453722122287133, 90533973988868134321542, 3559734642493103582865533, 146969730550281362048202174, 6356719089202681283092805137, 287440643937159436055153903598, 13562631117348347165659535163497 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

E.g.f. A(x) and related functions B(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. 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! +...

Related series.

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 A(x) + B(x) =  (A'(x) + B'(x)) / C(x),

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

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),

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

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),

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

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, n)}

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

CROSSREFS

Cf. A292120 (A+B+C), A292122 (B), A292123 (C), A292124 (A*B*C).

Sequence in context: A215755 A144535 A042641 * A236250 A042285 A041070

Adjacent sequences:  A292118 A292119 A292120 * A292122 A292123 A292124

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 08 2017

STATUS

approved

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Last modified July 22 02:20 EDT 2019. Contains 325210 sequences. (Running on oeis4.)