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A273955
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G.f. A(x) satisfies: A( x*A(x) - A(x)^2 ) = -x^3.
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2
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1, 1, -1, 2, -4, 12, -36, 115, -366, 1202, -4016, 13684, -47192, 164550, -578773, 2051994, -7324990, 26306860, -94980720, 344555898, -1255235744, 4590432218, -16845658552, 62014596756, -228956736888, 847546307549, -3145089430938, 11697191534690, -43595085475847, 162793363083734, -609005952973882, 2282129119421879, -8565364253229324, 32195552437196082, -121185775973925826, 456749227410641398
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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) x*A(x) - A(x)^2 = 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 = 3.9374997379511376037..., c = 0.034997955229443779... . - Vaclav Kotesovec, Jun 24 2016
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EXAMPLE
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G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 4*x^5 + 12*x^6 - 36*x^7 + 115*x^8 - 366*x^9 + 1202*x^10 - 4016*x^11 + 13684*x^12 - 47192*x^13 + 164550*x^14 +...
such that A( x*A(x) - A(x)^2 ) = -x^3.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 - x^4 + 2*x^5 - 3*x^6 + 12*x^7 - 36*x^8 + 118*x^9 - 366*x^10 + 1202*x^11 - 4004*x^12 + 13684*x^13 - 47192*x^14 + 164604*x^15 +...
x*A(x) - A(x)^2 = -x^3 - x^6 - 3*x^9 - 12*x^12 - 54*x^15 - 264*x^18 - 1362*x^21 - 7300*x^24 - 40245*x^27 - 226746*x^30 - 1299779*x^33 +...
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 + 54*x^5 - 264*x^6 + 1362*x^7 - 7300*x^8 + 40245*x^9 - 226746*x^10 + 1299779*x^11 - 7556310*x^12 + 44445150*x^13 - 264010326*x^14 + 1581537357*x^15 - 9543458802*x^16 + 57956158488*x^17 - 353941849554*x^18 +...
such that x*A(x) - A(x)^2 = 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, x*F - F^2), #A+1) ); A[n]}
for(n=1, 40, 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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