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A340338
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G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)^(n+1)).
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7
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1, 2, 4, 11, 35, 121, 444, 1689, 6592, 26258, 106313, 436203, 1809727, 7579202, 31999297, 136050472, 582002281, 2503242025, 10818689627, 46959246659, 204623676444, 894785832949, 3925297799901, 17270153317728, 76187650017660, 336934181461844
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OFFSET
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0,2
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 0..280
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FORMULA
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G.f. A(x) satisfies the following relations.
(1) A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)^(n+1)).
(2) A(x) = Sum_{n>=0} x^n*A(x)^n / (1 - x^(n+1)*A(x)^n).
(3) A(x) = Sum_{n>=0} x^(n^2+2*n) * A(x)^(n^2+n) * (1 - x^(2*n+2)*A(x)^(2*n+1))/((1 - x^(n+1)*A(x)^n)*(1 - x^(n+1)*A(x)^(n+1))).
a(n) ~ c * d^n / n^(3/2), where d = 4.6940696906229278555829659... and c = 0.71283897646430285525... - Vaclav Kotesovec, Jan 07 2021
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 4*x^2 + 11*x^3 + 35*x^4 + 121*x^5 + 444*x^6 + 1689*x^7 + 6592*x^8 + 26258*x^9 + 106313*x^10 + ...
where
A(x) = 1/(1 - x*A(x)) + x/(1 - x^2*A(x)^2) + x^2/(1 - x^3*A(x)^3) + x^3/(1 - x^4*A(x)^4) + x^4/(1 - x^5*A(x)^5) + ...
also
A(x) = 1/(1 - x) + x*A(x)/(1 - x^2*A(x)) + x^2*A(x)^2/(1 - x^3*A(x)^2) + x^3*A(x)^3/(1 - x^4*A(x)^3) + x^4*A(x)^4/(1 - x^5*A(x)^4) + ...
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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^m / (1 - x^(m+1)*A^(m+1) +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^m / (1 - x^(m+1)*A^m +x*O(x^n)) )); ; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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Sequence in context: A071794 A234006 A285002 * A107378 A186998 A243788
Adjacent sequences: A340335 A340336 A340337 * A340339 A340340 A340341
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jan 05 2021
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STATUS
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approved
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