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A371713
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Expansion of g.f. A(x) satisfying A(x)^2 = A( A(x)*(x + A(x)^2) ).
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5
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1, 1, 3, 10, 39, 161, 698, 3126, 14361, 67287, 320319, 1544894, 7532756, 37070678, 183892128, 918539002, 4615979653, 23321497085, 118391352149, 603585987830, 3089089467145, 15864868600157, 81737410659710, 422342729686590, 2188088882282654, 11363944086758244, 59152933495794684
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x*A(x) + A(x)^3 ).
(2) A(x)^4 = A( x*A(x)^3 + A(x)^5 + A(x)^6 ).
(3) A(x)^8 = A( x*A(x)^7 + A(x)^9 + A(x)^10 + A(x)^12 ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1) + Sum_{k=0..n-1} A(x)^(2^n+2^k) ) for n > 0.
(5) A(x) = x + Sum_{n>=0} A(x)^(2^n+1).
(6) A(x) = Series_Reversion(x - x*Sum_{n>=0} x^(2^n) ).
a(n) ~ c * d^n / n^(3/2), where d = 5.51142100999137014688261137378225123402050823381269982231021216596989145... and c = 0.07924552169373639393012621342284829291839319195254975892205166214809... - Vaclav Kotesovec, Apr 05 2024
The radius of convergence r = 0.1814414101530... = 1/d (d is given above) and A(r) satisfy: 1 = Sum_{n>=0} (2^n+1) * A(r)^(2^n) and r = A(r) - Sum_{n>=0} A(r)^(2^n+1), where A(r) = 0.319865507392391473797021103685180915354570766210154873070... - Paul D. Hanna, Apr 05 2024
c = sqrt(r) / sqrt(2*Pi * Sum_{k>=0} 2^k * (1 + 2^k) * A(r)^(2^k - 1)). - Vaclav Kotesovec, Apr 05 2024
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EXAMPLE
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G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 161*x^6 + 698*x^7 + 3126*x^8 + 14361*x^9 + 67287*x^10 + 320319*x^11 + 1544894*x^12 + ...
where A(x)^2 = A( A(x)*(x + A(x)^2) ).
RELATED SERIES.
(1) A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 107*x^6 + 460*x^7 + 2052*x^8 + 9394*x^9 + 43903*x^10 + 208570*x^11 + 1004263*x^12 + ...
(2) A(x)*(x + A(x)^2) = x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 88*x^6 + 374*x^7 + 1652*x^8 + 7512*x^9 + 34920*x^10 + 165198*x^11 + 792700*x^12 + ...
(3) Let R(x) be the series reversion of A(x), R(A(x)) = x, then
R(x) = x - x^2 - x^3 - x^5 - x^9 - x^17 - x^33 - x^65 + ... + -x^(2^n+1) + ...
and R(x) = R(x^2)/x - x^2.
SPECIFIC VALUES.
A(1/6) = 0.2367013365733826841498068726305704943941...
A(1/7) = 0.1823951399847440022737563157206822905959...
A(1/8) = 0.1515149787834965771672802816841610180120...
A(1/9) = 0.1303567976332909027691102900878848253626...
A(1/6)^2 = A(t) at t = A(1/6)*(1/6 + A(1/6)^2) = 0.05271201227864865...
A(1/7)^2 = A(t) at t = A(1/7)*(1/7 + A(1/7)^2) = 0.03212436773155026...
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MATHEMATICA
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nmax = 30; A[_] = 0; Do[A[x_] = x + Sum[A[x]^(2^k + 1), {k, 0, Log[nmax]/Log[2]}] + O[x]^(nmax + 1) // Normal, nmax + 1]; Rest[CoefficientList[A[x], x]] (* Vaclav Kotesovec, Apr 05 2024 *)
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PROG
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(PARI) /* G.f. Series_Reversion(x - x*Sum_{n>=0} x^(2^n)) */
{a(n) = my(A = serreverse(x - x*sum(k=0, #binary(n), x^(2^k)) +x*O(x^n))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1], F); for(i=1, n, A=concat(A, 0); F=x*Ser(A);
A[#A] = -polcoeff( F^2 - subst(F, x, F*(x + F^2)), #A+1) ); A[n]}
for(n=1, 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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