A292123 E.g.f. C(x) satisfies: C'(x) = A(x)*B(x) such that C(x)^2 - A(x)^2 = 8 and C(x)^2 - B(x)^2 = 5, where A(x) and B(x) are the e.g.f.s of A292121 and A292122, respectively.

3, 2, 15, 82, 759, 6698, 83355, 1111018, 17804811, 311922962, 6167999175, 132938646082, 3142313821119, 80223941595578, 2209482997765395, 65134627503574618, 2049303263312162451, 68484112629314107682, 2423689374810702823935, 90531454062326770264882, 3559791311479931284873479, 146968393568442835837277258, 6356752152236511568747181835, 287439789050158514775171177418, 13562654176000911126785202632091

Offset

0,1

Links

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

Formula

E.g.f. C(x) and related functions A(x) and B(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. 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! +...
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 C(x)^2 - A(x)^2 = 8.
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 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(C, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A292120 (A+B+C), A292121 (A), A292122 (B), A292124 (A*B*C).

Sequence in context: A190961 A346059 A126323 * A084886 A275463 A338280

Adjacent sequences: A292120 A292121 A292122 * A292124 A292125 A292126

Keyword

nonn

Author

Paul D. Hanna, Sep 08 2017

Status

approved