A278748 - OEIS

A278748

E.g.f. C(x) is defined by: C(x) = 2 + Integral A(x)*B(x)^2 - A(x)^2*B(x) dx, where A(x) and B(x) are described by A278746 and A278747, respectively.

2, 0, 2, -8, -90, 520, -566, 50904, 396710, -18390200, 131511978, -3005766632, 33380803270, 1757150970120, -44857682845174, 950900072932696, -28165020555314970, -73025855982929720, 24480067437941915242, -834839712703258127208, 34458996334508149067270, -773502135061735864720760, -12427737319809527785823286, 1295375801958445816867049944, -74746823183505251943756410330, 3392808342679593497307739258440

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OFFSET

0,1

LINKS

Table of n, a(n) for n=0..25.

FORMULA

Given e.g.f. C(x), functions A = A(x), B = B(x), and C = C(x) satisfy:

(1) A^2 + B^2 + C^2 = 5,

(2) A^3 + B^3 + C^3 = 9.

(3) A^4 + B^4 + C^4 = (A'^2 + B'^2 + C'^2 + 81)/5.

In vector form, functions A, B, and C may be defined by

(4) [A,B,C] = [0,1,2] + Integral [A,B,C] X [A^2,B^2,C^2] dx.

Specifically,

(4.a) A = Integral B*C^2 - B^2*C dx,

(4.b) B = 1 + Integral C*A^2 - C^2*A dx,

(4.c) C = 2 + Integral A*B^2 - A^2*B dx.

EXAMPLE

E.g.f.: C(x) = 2 + 2*x^2/2! - 8*x^3/3! - 90*x^4/4! + 520*x^5/5! - 566*x^6/6! + 50904*x^7/7! + 396710*x^8/8! - 18390200*x^9/9! + 131511978*x^10/10! - 3005766632*x^11/11! + 33380803270*x^12/12! + 1757150970120*x^13/13! - 44857682845174*x^14/14! + 950900072932696*x^15/15! +...

such that C(x) = 2 + Integral A(x)*B(x)^2 - A(x)^2*B(x) dx, and A(x) and B(x) begin as follows.

A(x) = 2*x + 6*x^3/3! - 24*x^4/4! - 1206*x^5/5! + 8280*x^6/6! - 52858*x^7/7! + 938952*x^8/8! + 6413418*x^9/9! - 322109160*x^10/10! + 4068033894*x^11/11! - 98801788536*x^12/12! + 1090129290506*x^13/13! + 34232660705880*x^14/14! - 1166338051942842*x^15/15! +...

B(x) = 1 - 8*x^2/2! + 16*x^3/3! - 72*x^4/4! + 640*x^5/5! + 6104*x^6/6! - 111888*x^7/7! + 1030936*x^8/8! - 20710880*x^9/9! + 100278648*x^10/10! + 4259777104*x^11/11! - 94939798408*x^12/12! + 2751909451200*x^13/13! - 58236639534824*x^14/14! - 39216365862992*x^15/15! +...

These series satisfy:

(1) A(x)^2 + B(x)^2 + C(x)^2 = 5,

(2) A(x)^3 + B(x)^3 + C(x)^3 = 9.

PROG

(PARI) {a(n) = my(A=x, B=1, C=2); for(i=1, n,

A = intformal(B*C^2 - B^2*C +x*O(x^n));

B = 1 + intformal(C*A^2 - C^2*A);

C = 2 + intformal(A*B^2 - A^2*B); ); n!*polcoeff(C, n)}

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

CROSSREFS

Cf. A278746, A278747.

Sequence in context: A196504 A182550 A004514 * A342286 A211930 A212026

Adjacent sequences: A278745 A278746 A278747 * A278749 A278750 A278751

KEYWORD

sign

AUTHOR

Paul D. Hanna, Dec 19 2016

STATUS

approved