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A279843
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E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral 2*A(x) dx.
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12
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1, 3, 18, 189, 2907, 59373, 1520019, 46964934, 1705072680, 71304998301, 3382510434561, 179805942701262, 10604941134768027, 688310036217216666, 48823571643364894410, 3762575719966561217301, 313432935903428395412205, 28098727418570995251538128, 2700377607104440375587008499, 277246288187233901613660728700
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f. A(x) satisfies:
(1) A(x - Integral A(x) dx) = x + Integral 2*A(x) dx.
(2) A(x) = x + 3 * G( (A(x) + 2*x)/3 ), where G(x) = Integral A(x) dx.
(3) A(x) = -2*x + 3 * Series_Reversion(x - Integral A(x) dx).
(4) A( (A(x) + 2*x)/3 ) = (A'(x) - 1)/(A'(x) + 2).
(5) A'(x - Integral A(x) dx) = (1 + 2*A(x))/(1 - A(x)).
a(n) = Sum_{k=0..n-1} A277410(n,k) * 3^(n-k-1).
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EXAMPLE
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E.g.f.: A(x) = x + 3*x^2/2! + 18*x^3/3! + 189*x^4/4! + 2907*x^5/5! + 59373*x^6/6! + 1520019*x^7/7! + 46964934*x^8/8! + 1705072680*x^9/9! + 71304998301*x^10/10! + 3382510434561*x^11/11! + 179805942701262*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - G(x)) = x + 2*G(x) where
G(x) = x^2/2! + 3*x^3/3! + 18*x^4/4! + 189*x^5/5! + 2907*x^6/6! + 59373*x^7/7! + 1520019*x^8/8! + 46964934*x^9/9! + 1705072680*x^10/10! +...
Also, A(x) = x + 3 * G( (A(x) + 2*x)/3 ).
RELATED SERIES.
We have (A(x) + 2*x)/3 = Series_Reversion( x - Integral A(x) dx ), where
(A(x) + 2*x)/3 = x + x^2/2! + 6*x^3/3! + 63*x^4/4! + 969*x^5/5! + 19791*x^6/6! + 506673*x^7/7! + 15654978*x^8/8! + 568357560*x^9/9! + 23768332767*x^10/10! + 1127503478187*x^11/11! + 59935314233754*x^12/12! +...
Further, A( (A(x) + 2*x)/3 ) = (A'(x) - 1)/(A'(x) + 2), which begins
A( (A(x) + 2*x)/3 ) = x + 4*x^2/2! + 33*x^3/3! + 441*x^4/4! + 8241*x^5/5! + 199071*x^6/6! + 5922360*x^7/7! + 209986506*x^8/8! + 8665824933*x^9/9! + 408861881955*x^10/10! + 21747689650404*x^11/11! +...
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PROG
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(PARI) /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
{a(n, p=1, q=2) = my(A=x, G); for(i=1, n, G = intformal(A +x*O(x^n)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q)) +x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n, 1, 2), ", "))
(PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=1, q=2) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); G=intformal(F); A[#A] = -polcoeff(subst(F, x, x - p*G) - q*G, #A) ); n!*A[n]}
for(n=1, 30, print1(a(n, 1, 2), ", "))
(PARI) /* Informal code to generate the first N terms: */
{N=20; p=1; q=2; A=x; for(i=1, N, G=intformal(A +x*O(x^N)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q))); Vec(serlaplace(A))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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