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A372578
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Expansion of g.f. A(x) satisfying A( x*A(x) + 2*A(x)^3 ) = A(x)^2.
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2
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1, 2, 10, 60, 406, 2940, 22304, 174960, 1407582, 11550396, 96299472, 813433712, 6946442776, 59872428672, 520174647424, 4550665293920, 40052871669422, 354421196057404, 3151211548631856, 28137903707808048, 252219507331523688, 2268719274696321856, 20472066335198022080, 185268984285773695200
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OFFSET
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1,2
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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) + 2*A(x)^3 ).
(2) A(x)^4 = A( x*A(x)^3 + 2*A(x)^5 + 2*A(x)^6 ).
(3) A(x)^8 = A( x*A(x)^7 + 2*A(x)^9 + 2*A(x)^10 + 2*A(x)^12 ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1) + 2*Sum_{k=0,n-1} A(x)^(2^n+2^k) ).
(5) A(x) = x + 2*Sum_{n>=0} A(x)^(2^n+1).
(6) A(x) = Series_Reversion( x - 2*x*Sum_{n>=0} x^(2^n) ).
The radius of convergence r and A(r) satisfy: 1 = Sum_{n>=0} 2*(2^n+1) * A(r)^(2^n) and r = A(r) - 2*Sum_{n>=0} A(r)^(2^n+1), where r = 0.103594274393575546296984777950632418580281502255382627835... and A(r) = 0.191573759982214348953869719011237665707785580853712880852...
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EXAMPLE
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G.f.: A(x) = x + 2*x^2 + 10*x^3 + 60*x^4 + 406*x^5 + 2940*x^6 + 22304*x^7 + 174960*x^8 + 1407582*x^9 + 11550396*x^10 + 96299472*x^11 + 813433712*x^12 + ...
where A( x*A(x) + 2*A(x)^3 ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 160*x^5 + 1152*x^6 + 8704*x^7 + 68088*x^8 + 546656*x^9 + 4478720*x^10 + 37294080*x^11 + ...
A(x)^3 = x^3 + 6*x^4 + 42*x^5 + 308*x^6 + 2358*x^7 + 18612*x^8 + 150424*x^9 + 1238688*x^10 + 10355982*x^11 + 87672468*x^12 + ...
The series reversion R(x) of A(x), R(A(x)) = x, begins:
R(x) = x - 2*x^2 - 2*x^3 - 2*x^5 - 2*x^9 - 2*x^17 - 2*x^33 - 2*x^65 - 2*x^129 - 2*x^257 - 2*x^513 + ... + -2*x^(2^n+1) + ...
SPECIFIC VALUES.
A(1/10) = 0.1580645870348513671680526916072548213169829162556439...
A(1/11) = 0.1278454819475039498675733418966788971517121949516108...
A(1/12) = 0.1104694875320629136831876267359845627848091250498995...
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PROG
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(PARI) {a(n) = my(A=serreverse(x - 2*x*sum(m=0, #binary(n), x^(2^m) +x*O(x^n)))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0); F=Ser(A); A[#A] = polcoeff( subst(F, x, x*F + 2*F^3) - F^2, #A) ); A[n+1]}
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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