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A273095
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G.f. A(x) satisfies: A(x)^2 - 2*A(x)^3 = A(x^2).
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6
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1, 1, 3, 9, 34, 131, 544, 2321, 10219, 45858, 209422, 969115, 4536240, 21435168, 102120900, 489966937, 2365411849, 11481936107, 56005441585, 274365808690, 1349355995822, 6659759051278, 32975144338094, 163753271207603, 815384381782875, 4070152108372400, 20363490114853946, 102097178305283104, 512896449689275282, 2581316849641991364, 13013547947445475296, 65712064033864711137, 332312118095745844303, 1682904769274022037513
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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) satisfies: A(B(x)^2) = x^2 - 2*x^3, where A(B(x)) = x.
a(n) ~ c * d^n / n^(3/2), where d = 5.29878119718866901105709936425... and c = 0.0784096654417593202431027... - Vaclav Kotesovec, Sep 02 2017
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EXAMPLE
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G.f.: A(x) = x + x^2 + 3*x^3 + 9*x^4 + 34*x^5 + 131*x^6 + 544*x^7 + 2321*x^8 + 10219*x^9 + 45858*x^10 + 209422*x^11 + 969115*x^12 + 4536240*x^13 +...
where A(x)^2 - 2*A(x)^3 = A(x^2).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 24*x^5 + 95*x^6 + 384*x^7 + 1635*x^8 + 7128*x^9 + 31858*x^10 + 144780*x^11 + 667805*x^12 + 3116520*x^13 + 14691616*x^14 +...
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 46*x^6 + 192*x^7 + 813*x^8 + 3564*x^9 + 15912*x^10 + 72390*x^11 + 333837*x^12 + 1558260*x^13 + 7345536*x^14 + 34924036*x^15 +...
where
A(x)^2 - 2*A(x)^3 = x^2 + x^4 + 3*x^6 + 9*x^8 + 34*x^10 + 131*x^12 + 544*x^14 +...
Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
B(x) = x - x^2 - x^3 + x^4 - 2*x^5 + 2*x^6 - 6*x^7 - 2*x^8 - x^9 + x^10 - 43*x^11 + 35*x^12 - 48*x^13 - 64*x^14 - 256*x^15 + 552*x^16 - 1791*x^17 + 1583*x^18 - 3941*x^19 + 3149*x^20 - 12464*x^21 + 8696*x^22 - 36452*x^23 + 30772*x^24 +...
where B(x^2 - 2*x^3) = B(x)^2.
Further, we have
A(B(x)^2) = C(x) = x^2 - 2*x^3,
A(B(x)^4) = C(C(x)) = x^4 - 4*x^5 + 2*x^6 + 12*x^7 - 24*x^8 + 16*x^9,
A(B(x)^8) = C(C(C(x))) = x^8 - 8*x^9 + 20*x^10 + 8*x^11 - 142*x^12 + 296*x^13 - 188*x^14 - 360*x^15 + 1464*x^16 - 3360*x^17 + 4176*x^18 + 2400*x^19 - 19200*x^20 + 30720*x^21 - 10752*x^22 - 39936*x^23 + 79872*x^24 - 73728*x^25 + 36864*x^26 - 8192*x^27, ...
so that A(B(x)^(2^n)) = C^n(x), the n-th iteration of C(x) = x^2 - 2*x^3.
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PROG
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(PARI) /* From A(B(x)^2) = x^2 - 2*x^3, where A(B(x)) = x: */
{a(n) = my(A=[1, 1], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(F, x, B^2))[#A]/2); A[n]}
for(n=1, 40, 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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