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A251180
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G.f. satisfies: A(x) = Sum_{n>=0} (A(x)^n + 2)^n * x^n / (1+2*x)^(n+1).
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4
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1, 1, 2, 11, 59, 376, 2566, 18646, 141857, 1120851, 9141387, 76635239, 658411100, 5784858465, 51899580702, 474971067333, 4431203311040, 42128438013171, 408111843546201, 4028707682556147, 40534978365189110, 415825232653264747, 4350847058443120856, 46450772334813948748
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OFFSET
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0,3
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COMMENTS
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Compare to: F(x) = Sum_{n>=0} (F(x)^n + 2)^n * x^n / (1+3*x)^(n+1) holds when F(x) = 1.
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LINKS
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FORMULA
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G.f. satisfies:
(1) A(x) = Sum_{n>=0} A(x)^(n^2) * x^n / (1+2*x - 2*x*A(x)^n)^(n+1).
(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * (A(x)^k + 2)^k.
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 59*x^4 + 376*x^5 + 2566*x^6 +...
where we have the identities:
(0) A(x) = 1/(1+2*x) + (A(x)+2)*x/(1+2*x)^2 + (A(x)^2+2)^2*x^2/(1+2*x)^3 + (A(x)^3+2)^3*x^3/(1+2*x)^4 + (A(x)^4+2)^4*x^4/(1+2*x)^5 + (A(x)^5+2)^5*x^5/(1+2*x)^6 +...
(1) A(x) = 1 + A(x)*x/(1+2*x - 2*x*A(x))^2 + A(x)^4*x^2/(1+2*x - 2*x*A(x)^2)^3 + A(x)^9*x^3/(1+2*x - 2*x*A(x)^3)^4 + A(x)^16*x^4/(1+2*x - 2*x*A(x)^4)^5 + A(x)^25*x^5/(1+2*x - 2*x*A(x)^5)^6 + A(x)^36*x^6/(1+2*x - 2*x*A(x)^6)^7 +...
(2) A(x) = 1 - x*(2 - (A(x)+2)) + x^2*(2^2 - 2*2*(A(x)+2) + (A(x)^2+2)^2) - x^3*(2^3 - 3*2^2*(A(x)+2) + 3*2*(A(x)^2+2)^2 - (A(x)^3+2)^3) + x^4*(2^4 - 4*2^3*(A(x)+2) + 6*2^2*(A(x)^2+2)^2 - 4*2*(A(x)^3+2)^3 + (A(x)^4+2)^4) - x^5*(2^5 - 5*2^4*(A(x)+2) + 10*2^3*(A(x)^2+2)^2 - 10*2^2*(A(x)^3+2)^3 + 5*2*(A(x)^4+2)^4 - (A(x)^5+2)^5) +...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (A^m + 2)^m * x^m / (1+2*x +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, A^(m^2) * x^m / (1+2*x - 2*x*A^m +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m * sum(k=0, m, binomial(m, k) * (-2)^(m-k) * (A^k + 2)^k +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, 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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