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A302105
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G.f. A(x) satisfies: A(x) = Sum_{n>=0} (4 + x*A(x)^n)^n / 5^(n+1).
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4
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1, 1, 10, 175, 3835, 95090, 2551480, 72360700, 2139052845, 65329175385, 2049247480265, 65752776679275, 2151923601749290, 71691421965972905, 2428004656549037580, 83523871228996755395, 2917260885363111908770, 103451501815230690971935, 3726040763307222530311125, 136400452641372633368206185, 5080478361492407723101242440
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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} (4 + x*G(x)^k)^n / 5^(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} (4 + x*A(x)^n)^n / 5^(n+1).
(2) A(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (5 - 4*A(x)^n)^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + x + 10*x^2 + 175*x^3 + 3835*x^4 + 95090*x^5 + 2551480*x^6 + 72360700*x^7 + 2139052845*x^8 + 65329175385*x^9 + 2049247480265*x^10 + ...
such that
A(x) = 4/5 + (4 + x*A(x))/5^2 + (4 + x*A(x)^2)^2/5^3 + (4 + x*A(x)^3)^3/5^4 + (4 + x*A(x)^4)^4/5^5 + (4 + x*A(x)^5)^5/5^6 + (4 + x*A(x)^6)^6/5^7 + ...
Also, due to a series identity,
A(x) = 1 + x*A(x)/(5 - 4*A(x))^2 + x^2*A(x)^4/(5 - 4*A(x)^2)^3 + x^3*A(x)^9/(5 - 4*A(x)^3)^4 + x^4*A(x)^16/(5 - 4*A(x)^4)^5 + x^5*A(x)^25/(5 - 4*A(x)^5)^6 + x^6*A(x)^36/(5 - 4*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (5 - 4*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) / (5 - 4*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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