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A276364
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G.f. A(x) satisfies: A(x - A(x)^3) = x + A(x)^3, where A(x) = Sum_{n>=1} a(n)*x^(2*n-1).
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13
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1, 2, 18, 252, 4410, 88734, 1969668, 47104056, 1195658550, 31891944750, 887565934494, 25639389304560, 765765781572600, 23574635888791804, 746297727831434376, 24247096863466015152, 807243935471150901066, 27503153109167182217082, 957899411829034037383374, 34073454839478198669105444, 1236879534288183156526996062, 45788365378826408823663436974, 1727576456033196960394178300184
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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) also satisfies:
(1) A(x) = x + 2 * A( x/2 + A(x)/2 )^3.
(2) A(x) = -x + 2 * Series_Reversion(x - A(x)^3).
(3) R(x) = -x + 2 * Series_Reversion(x + A(x)^3), where R(A(x)) = x.
(4) R( ( x/2 - R(x)/2 )^(1/3) ) = x/2 + R(x)/2, where R(A(x)) = x.
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EXAMPLE
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G.f.: A(x) = x + 2*x^3 + 18*x^5 + 252*x^7 + 4410*x^9 + 88734*x^11 + 1969668*x^13 + 47104056*x^15 + 1195658550*x^17 + 31891944750*x^19 + 887565934494*x^21 + 25639389304560*x^23 + 765765781572600*x^25 +...
such that A(x - A(x)^3) = x + A(x)^3.
RELATED SERIES.
Note that Series_Reversion(x - A(x)^3) = x/2 + A(x)/2, which begins:
Series_Reversion(x - A(x)^2) = x + x^3 + 9*x^5 + 126*x^7 + 2205*x^9 + 44367*x^11 + 984834*x^13 + 23552028*x^15 + 597829275*x^17 + 15945972375*x^19 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 2*x^3 - 6*x^5 - 60*x^7 - 830*x^9 - 13950*x^11 - 267156*x^13 - 5629752*x^15 - 127807290*x^17 - 3082830030*x^19 - 78254901810*x^21 - 2076067799280*x^23 - 57266880966792*x^25 +...
then Series_Reversion(x + A(x)^3) = x/2 + R(x)/2.
Also, A(x) = x + 2 * A( x/2 + A(x)/2 )^3, where
A( x/2 + A(x)/2 ) = x + 3*x^3 + 33*x^5 + 528*x^7 + 10235*x^9 + 224001*x^11 + 5343738*x^13 + 136167888*x^15 + 3659113701*x^17 + 102800460825*x^19 + 3001057504233*x^21 + 90627712970220*x^23 + 2821487673544920*x^25 +...
and
A( x/2 + A(x)/2 )^3 = x^3 + 9*x^5 + 126*x^7 + 2205*x^9 + 44367*x^11 + 984834*x^13 + 23552028*x^15 + 597829275*x^17 + 15945972375*x^19 +...
which equals -x/2 + A(x)/2.
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MATHEMATICA
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nmin = 1; nmax = 60; sol = {b[1] -> 1}; nsol = Length[sol];
Do[A[x_] = Sum[b[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x - A[x]^3] - x - A[x]^3 + O[x]^(n+1), x][[nsol+1;; ]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nsol + 1, nmax}];
a[n_] := b[2n-1];
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PROG
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(PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, [0, 0]); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - F^3) - F^3, #A) ); A[2*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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