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A323568 E.g.f. exp( Integral B(x) dx ) = (A(x) + C(x))/8 where A(x) = 3 + Integral B(x)*C(x) dx and C(x) = 5 + Integral A(x)*B(x) dx such that A(x)^2 - 3^2 = B(x)^2 - 4^2 = C(x)^2 - 5^2. 7
1, 4, 31, 380, 6017, 119812, 2863903, 79849340, 2544494081, 91218404356, 3633458059039, 159202837770620, 7609742731484033, 394049923598413828, 21974409170191973407, 1312944562859473313660, 83676651905307684333569, 5666190077265731685365764, 406257664391186624860008991, 30746316355830218639758848380, 2449407313095934366223281388417, 204888804298166571602163435568132 (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. exp( Integral B(x) dx ) where related series A(x), B(x), and C(x) satisfy the following relations.

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

(1b) B(x) = 4 + Integral A(x)*C(x) dx.

(1c) C(x) = 5 + Integral A(x)*B(x) dx.

(2a) C(x)^2 - B(x)^2 = 9.

(2b) C(x)^2 - A(x)^2 = 16.

(2c) B(x)^2 - A(x)^2 = 7.

(3a) A(x)*B(x)*C(x) = A(x)*A'(x) = B(x)*B'(x) = C(x)*C'(x).

(3b) Integral 2*A(x)*B(x)*C(x) dx = A(x)^2 - 9 = B(x)^2 - 16 = C(x)^2 - 25.

(4a) B(x) + C(x) = 9 * exp( Integral A(x) dx ).

(4b) A(x) + C(x) = 8 * exp( Integral B(x) dx ).

(4c) A(x) + B(x) = 7 * exp( Integral C(x) dx ).

EXAMPLE

E.g.f. exp( Integral B(x) dx ) = 1 + 4*x + 31*x^2/2! + 380*x^3/3! + 6017*x^4/4! + 119812*x^5/5! + 2863903*x^6/6! + 79849340*x^7/7! + 2544494081*x^8/8! + 91218404356*x^9/9! + 3633458059039*x^10/10! + ...

RELATED SERIES.

A(x) = 3 + 20*x + 123*x^2/2! + 1540*x^3/3! + 23871*x^4/4! + 480260*x^5/5! + 11449599*x^6/6! + 319491220*x^7/7! + 10176946203*x^8/8! + 364884459380*x^9/9! + 14533663841187*x^10/10! + ... + A323563(n)*x^n/n! + ...

such that A(x) = 3 + Integral B(x)*C(x) dx.

B(x) = 4 + 15*x + 136*x^2/2! + 1470*x^3/3! + 24128*x^4/4! + 478320*x^5/5! + 11464768*x^6/6! + 319326960*x^7/7! + 10178837504*x^8/8! + 364859900160*x^9/9! + 14534008182784*x^10/10! + ... + A323564(n)*x^n/n! + ...

such that B(x) = 4 + Integral A(x)*C(x) dx.

C(x) = 5 + 12*x + 125*x^2/2! + 1500*x^3/3! + 24265*x^4/4! + 478236*x^5/5! + 11461625*x^6/6! + 319303500*x^7/7! + 10179006445*x^8/8! + 364862775468*x^9/9! + 14534000631125*x^10/10! + ... + A323565(n)*x^n/n! + ...

such that C(x) = 5 + Integral A(x)*B(x) dx.

A(x)^2 = 9 + 120*x + 1538*x^2/2! + 24000*x^3/3! + 480400*x^4/4! + 11444160*x^5/5! + 319475984*x^6/6! + 10177152000*x^7/7! + 364886675200*x^8/8! + 14533662074880*x^9/9! + 636813851059712*x^10/10! ...

such that C(x)^2 - A(x)^2 = 16 and B(x)^2 - A(x)^2 = 7.

A(x) + B(x) = 7 * exp( Integral C(x) dx ) = 7 + 35*x + 259*x^2/2! + 3010*x^3/3! + 47999*x^4/4! + 958580*x^5/5! + 22914367*x^6/6! + 638818180*x^7/7! + 20355783707*x^8/8! + 729744359540*x^9/9! + 29067672023971*x^10/10! + ...

A(x) + C(x) = 8 * exp( Integral B(x) dx ) = 8 + 32*x + 248*x^2/2! + 3040*x^3/3! + 48136*x^4/4! + 958496*x^5/5! + 22911224*x^6/6! + 638794720*x^7/7! + 20355952648*x^8/8! + 729747234848*x^9/9! + 29067664472312*x^10/10! + ...

B(x) + C(x) = 9 * exp( Integral A(x) dx ) = 9 + 27*x + 261*x^2/2! + 2970*x^3/3! + 48393*x^4/4! + 956556*x^5/5! + 22926393*x^6/6! + 638630460*x^7/7! + 20357843949*x^8/8! + 729722675628*x^9/9! + 29068008813909*x^10/10! + ...

exp( Integral A(x) dx ) = 1 + 3*x + 29*x^2/2! + 330*x^3/3! + 5377*x^4/4! + 106284*x^5/5! + 2547377*x^6/6! + 70958940*x^7/7! + 2261982661*x^8/8! + 81080297292*x^9/9! + 3229778757101*x^10/10! + ... + A323569(n)*x^n/n! + ...

exp( Integral C(x) dx ) = 1 + 5*x + 37*x^2/2! + 430*x^3/3! + 6857*x^4/4! + 136940*x^5/5! + 3273481*x^6/6! + 91259740*x^7/7! + 2907969101*x^8/8! + 104249194220*x^9/9! + 4152524574853*x^10/10! + ... + A323567(n)*x^n/n! + ...

A(x)*B(x)*C(x) = 60 + 769*x + 12000*x^2/2! + 240200*x^3/3! + 5722080*x^4/4! + 159737992*x^5/5! + 5088576000*x^6/6! + 182443337600*x^7/7! + 7266831037440*x^8/8! + 318406925529856*x^9/9! + 15219462171648000*x^10/10! + ... + A323566(n)*x^n/n! + ...

such that A(x)*B(x)*C(x) = A(x)*A'(x) = B(x)*B'(x) = C(x)*C'(x).

PROG

(PARI) {ac8(n) = my(A=3, B=4, C=5); for(i=1, n,

A = 3 + intformal(B*C +x*O(x^n));

B = 4 + intformal(A*C);

C = 5 + intformal(A*B); );

n! * polcoeff( (A+C)/8 , n)}

for(n=0, 30, print1(ac8(n), ", "))

CROSSREFS

Cf. A323563 (A), A323564 (B), A323565 (C), A323566 (A*B*C), A323567 ((A+B)/7), A323569 ((B+C)/9).

Sequence in context: A145087 A215529 A005046 * A174324 A211194 A237581

Adjacent sequences:  A323565 A323566 A323567 * A323569 A323570 A323571

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 18 2019

STATUS

approved

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Last modified May 19 10:36 EDT 2019. Contains 323390 sequences. (Running on oeis4.)