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A340355
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G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^(2*n)*A(x)^n / (1 - x^(n+1)*A(x)^(n+2)).
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10
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1, 1, 4, 15, 73, 373, 2036, 11518, 67108, 399596, 2421477, 14883232, 92561649, 581401130, 3683031612, 23502839520, 150944260610, 974905750378, 6328238976958, 41261578774953, 270119042203681, 1774773646080126, 11699419572947070, 77355905980770122, 512885661648043804
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OFFSET
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0,3
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COMMENTS
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The g.f. A(x) of this sequence is motivated by the following identity:
Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;
here, p = x^2*A(x), q = x*A(x)^2, and r = x*A(x).
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LINKS
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FORMULA
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G.f. A(x) satisfies the following relations.
(1) A(x) = Sum_{n>=0} x^(2*n) * A(x)^n / (1 - x^(n+1)*A(x)^(n+2)).
(2) A(x) = Sum_{n>=0} x^n * A(x)^(2*n) / (1 - x^(n+2)*A(x)^(n+1)).
(3) A(x) = Sum_{n>=0} x^(n^2+3*n) * A(x)^(n^2+3*n) * (1 - x^(2*n+3)*A(x)^(2*n+3)) / ((1 - x^(n+1)*A(x)^(n+2))*(1 - x^(n+2)*A(x)^(n+1))).
a(n) ~ c * d^n / n^(3/2), where d = 7.060918158410189777854181567407... and c = 0.2611318997628883837033125... - Vaclav Kotesovec, Jan 07 2021
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EXAMPLE
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G.f.: A(x) = 1 + x + 4*x^2 + 15*x^3 + 73*x^4 + 373*x^5 + 2036*x^6 + 11518*x^7 + 67108*x^8 + 399596*x^9 + 2421477*x^10 + ...
where
A(x) = 1/(1 - x*A(x)^2) + x^2*A(x)/(1 - x^2*A(x)^3) + x^4*A(x)^2/(1 - x^3*A(x)^4) + x^6*A(x)^3/(1 - x^4*A(x)^5) + x^8*A(x)^4/(1 - x^5*A(x)^6) + ...
also
A(x) = 1/(1 - x^2*A(x)) + x*A(x)^2/(1 - x^3*A(x)^2) + x^2*A(x)^4/(1 - x^4*A(x)^3) + x^3*A(x)^6/(1 - x^5*A(x)^4) + x^4*A(x)^8/(1 - x^6*A(x)^5) + ...
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PROG
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(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^(2*m)*A^m / (1 - x^(m+1)*A^(m+2) +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^m*A^(2*m) / (1 - x^(m+2)*A^(m+1) +x*O(x^n)) )); ; 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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