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A268039
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G.f. A(x) satisfies: A( x*A(x) - A(x)^3 ) = x^2.
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4
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1, 1, 2, 9, 40, 192, 959, 4988, 26587, 144672, 800229, 4486914, 25444355, 145681030, 840988369, 4889658927, 28607653570, 168299372745, 994978254267, 5908150711835, 35221346706997, 210723727773531, 1264832205228154, 7614504644529573, 45965528482747194, 278169935575048042, 1687298108487873673, 10256585388232231101, 62470753620679133927, 381201089984659788693, 2330136670337522460729, 14266260150998229954489
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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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a(n) ~ c * d^n / n^(3/2), where d = 6.426300389361325672327464898259648... and c = 0.035189970759375828095135204598637... . - Vaclav Kotesovec, May 03 2016
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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 9*x^4 + 40*x^5 + 192*x^6 + 959*x^7 + 4988*x^8 + 26587*x^9 + 144672*x^10 + 800229*x^11 + 4486914*x^12 +...
where A( x*A(x) - A(x)^3 ) = x^2.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 40*x^6 + 192*x^7 + 963*x^8 + 4988*x^9 + 26589*x^10 + 144672*x^11 + 800253*x^12 + 4486914*x^13 + 25444374*x^14 +...
x*A(x) - A(x)^3 = x^2 - x^4 - 4*x^8 - 2*x^10 - 24*x^12 - 19*x^14 - 206*x^16 - 194*x^18 - 1980*x^20 - 2390*x^22 - 20920*x^24 - 31626*x^26 - 236114*x^28 +...
Let B(x) be the series reversion of the g.f. A(x), so that A(B(x)) = x, then
B(x) = x - x^2 - 4*x^4 - 2*x^5 - 24*x^6 - 19*x^7 - 206*x^8 - 194*x^9 - 1980*x^10 - 2390*x^11 - 20920*x^12 - 31626*x^13 - 236114*x^14 +...
such that B(x^2) = x*A(x) - A(x)^3.
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PROG
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(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -Vec(subst(F, x, x*F - F^3))[#A] ); 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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