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A359723
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a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.
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4
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1, 1, 7, 28, 151, 803, 4108, 22532, 125449, 705929, 4035955, 23332364, 136111591, 800561116, 4741777880, 28258286033, 169322163149, 1019483819757, 6164900341534, 37425357962592, 228002416106605, 1393503512669230, 8541839907812651, 52500559705299795, 323483846045526418
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OFFSET
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0,3
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COMMENTS
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Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 3*x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*3^k, for n >= 0.
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EXAMPLE
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G.f.: A(x) = 1 + x + 7*x^2 + 28*x^3 + 151*x^4 + 803*x^5 + 4108*x^6 + 22532*x^7 + 125449*x^8 + 705929*x^9 + 4035955*x^10 + 23332364*x^11 + 136111591*x^12 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.150684304746792807618050217238804920801612774142866...
A(1/7) = 1.67848119643298635311797131334138331526984303696733717...
A(1/8) = 1.40389487408504106142147713148599989460789630965507028...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (3 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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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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