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A272411
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G.f. A(x) satisfies: A( A(x)^2 - x*A(x) ) = x^3.
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2
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1, 1, -1, 2, -6, 16, -48, 155, -506, 1706, -5888, 20608, -73152, 262672, -951929, 3478158, -12798568, 47384216, -176387016, 659776638, -2478574412, 9347514586, -35376839998, 134317287748, -511463365764, 1952816800973, -7474463834606, 28673987914262, -110233267218581, 424608422717362, -1638541384230970, 6333831090142919, -24522697340016084, 95086658516947002
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OFFSET
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1,4
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LINKS
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FORMULA
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If A(B(x)) = x, then g.f. A(x) and B(x) satisfy:
(1) A(x)^2 - x*A(x) = B(x^3).
(2) A(x) = x - x*C( -B(x^3)/x^2 ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 4.06250021724219826323934729... and c = 0.03554943075321525313806189... . - Vaclav Kotesovec, May 03 2016
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EXAMPLE
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G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 6*x^5 + 16*x^6 - 48*x^7 + 155*x^8 - 506*x^9 + 1706*x^10 - 5888*x^11 + 20608*x^12 - 73152*x^13 + 262672*x^14 +...
where A( A(x)^2 - x*A(x) ) = x^3.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 - x^4 + 2*x^5 - 7*x^6 + 16*x^7 - 48*x^8 + 158*x^9 - 506*x^10 + 1706*x^11 - 5900*x^12 + 20608*x^13 - 73152*x^14 +...
A(x)^2 - x*A(x) = x^3 - x^6 + 3*x^9 - 12*x^12 + 56*x^15 - 282*x^18 + 1494*x^21 - 8210*x^24 + 46365*x^27 - 267444*x^30 +...
Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then
B(x) = x - x^2 + 3*x^3 - 12*x^4 + 56*x^5 - 282*x^6 + 1494*x^7 - 8210*x^8 + 46365*x^9 - 267444*x^10 + 1568995*x^11 - 9332820*x^12 + 56156610*x^13 +...
such that A(x)^2 - x*A(x) = B(x^3).
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PROG
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(PARI) {a(n) = my(A=[1, 1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = polcoeff(x^3 - subst(F, x, F^2 - x*F), #A+1) ); A[n]}
for(n=1, 50, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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