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A302103
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G.f. A(x) satisfies: A(x) = Sum_{n>=0} (2 + x*A(x)^n)^n / 3^(n+1).
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4
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1, 1, 6, 63, 837, 12672, 208686, 3647568, 66697203, 1264307667, 24696153573, 495076265421, 10157438738790, 212900154037875, 4553735135491134, 99341289091151409, 2210262851488661562, 50173932628981325523, 1162965513498859292415, 27554435907912281877315, 668277970101220006626558, 16617278354076763108026795
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OFFSET
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0,3
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COMMENTS
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Compare to: G(x) = Sum_{n>=0} (2 + x*G(x)^k)^n / 3^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} (2 + x*A(x)^n)^n / 3^(n+1).
(2) A(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (3 - 2*A(x)^n)^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + x + 6*x^2 + 63*x^3 + 837*x^4 + 12672*x^5 + 208686*x^6 + 3647568*x^7 + 66697203*x^8 + 1264307667*x^9 + 24696153573*x^10 + ...
such that
A(x) = 2/3 + (2 + x*A(x))/3^2 + (2 + x*A(x)^2)^2/3^3 + (2 + x*A(x)^3)^3/3^4 + (2 + x*A(x)^4)^4/3^5 + (2 + x*A(x)^5)^5/3^6 + (2 + x*A(x)^6)^6/3^7 + ...
Also, due to a series identity,
A(x) = 1 + x*A(x)/(3 - 2*A(x))^2 + x^2*A(x)^4/(3 - 2*A(x)^2)^3 + x^3*A(x)^9/(3 - 2*A(x)^3)^4 + x^4*A(x)^16/(3 - 2*A(x)^4)^5 + x^5*A(x)^25/(3 - 2*A(x)^5)^6 + x^6*A(x)^36/(3 - 2*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (3 - 2*A(x)^n)^(n+1) + ...
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PROG
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(PARI) {a(n) = my(A=1); for(i=0, n, A = sum(m=0, n, x^m * A^(m^2) / (3 - 2*A^m + x*O(x^n))^(m+1) )); polcoeff(A, n)}
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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