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A251181
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G.f. satisfies: A(x) = Sum_{n>=0} (A(x)^n - 1)^n * x^n / (1-2*x)^(n+1).
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4
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1, 2, 6, 22, 102, 582, 3838, 28006, 220182, 1835942, 16090254, 147392806, 1406155462, 13938397190, 143329786526, 1527361429606, 16853799768310, 192460135069030, 2273147914341294, 27752645471624486, 350014361895293862, 4556667705994310342, 61182067015102861502, 846481772532061942374
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OFFSET
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0,2
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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 + 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 - 1)^k.
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 6*x^2 + 22*x^3 + 102*x^4 + 582*x^5 + 3838*x^6 +...
where we have the identities:
(0) A(x) = 1/(1-2*x) + (A(x)-1)*x/(1-2*x)^2 + (A(x)^2-1)^2*x^2/(1-2*x)^3 + (A(x)^3-1)^3*x^3/(1-2*x)^4 + (A(x)^4-1)^4*x^4/(1-2*x)^5 + (A(x)^5-1)^5*x^5/(1-2*x)^6 +...
(1) A(x) = 1/(1-x) + A(x)*x/(1-2*x + x*A(x))^2 + A(x)^4*x^2/(1-2*x + x*A(x)^2)^3 + A(x)^9*x^3/(1-2*x + x*A(x)^3)^4 + A(x)^16*x^4/(1-2*x + x*A(x)^4)^5 + A(x)^25*x^5/(1-2*x + x*A(x)^5)^6 + A(x)^36*x^6/(1-2*x + x*A(x)^6)^7 +...
(2) A(x) = 1 + x*(2 + (A(x)-1)) + x^2*(2^2 + 2*2*(A(x)-1) + (A(x)^2-1)^2) + x^3*(2^3 + 3*2^2*(A(x)-1) + 3*2*(A(x)^2-1)^2 + (A(x)^3-1)^3) + x^4*(2^4 + 4*2^3*(A(x)-1) + 6*2^2*(A(x)^2-1)^2 + 4*2*(A(x)^3-1)^3 + (A(x)^4-1)^4) + x^5*(2^5 + 5*2^4*(A(x)-1) + 10*2^3*(A(x)^2-1)^2 + 10*2^2*(A(x)^3-1)^3 + 5*2*(A(x)^4-1)^4 + (A(x)^5-1)^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 - 1)^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 + 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 - 1)^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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