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A228862
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G.f. satisfies: x = A(x - A(x^2 - A(x^3 - A(x^4 - A(x^5 -...))))).
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3
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1, 1, 1, 2, 5, 11, 24, 58, 146, 365, 922, 2383, 6243, 16463, 43748, 117224, 316157, 857088, 2334700, 6388017, 17546354, 48361208, 133710567, 370744754, 1030649811, 2871950293, 8020308614, 22443012438, 62919001546, 176699520967, 497039125163, 1400236234543, 3950262035542
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OFFSET
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1,4
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COMMENTS
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The g.f. of A228863 equals the series reversion of the g.f. of this sequence.
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^(3/2), where d = 2.9489373... and c = 0.27314... - Vaclav Kotesovec, Sep 03 2017
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 11*x^6 + 24*x^7 + 58*x^8 +...
Let G(x) be the series reversion of A(x) (cf. A228863), then
(1) G(x) = x - x^2 + x^3 - 2*x^4 + 3*x^5 - 4*x^6 + 6*x^7 - 10*x^8 + 18*x^9 - 35*x^10 + 71*x^11 - 147*x^12 + 303*x^13 - 616*x^14 + 1244*x^15 +...
(2) G(x - G(x)) = x^2 - x^3 + x^4 - x^5 + x^7 - x^8 + x^10 - x^11 + 3*x^13 - 10*x^14 + 17*x^15 - 14*x^16 - 6*x^17 + 38*x^18 +...
(3) G(x^2 - G(x - G(x))) = x^3 - x^4 + x^5 - x^6 + x^7 - 2*x^8 + 3*x^9 - 3*x^10 + 3*x^11 - 5*x^12 + 8*x^13 - 9*x^14 + 10*x^15 +...
(4) G(x^3 - G(x^2 - G(x - G(x)))) = x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^11 - x^12 + x^13 - 2*x^14 + 2*x^15 - x^17 + x^18 - 3*x^19 + 4*x^20 +...
(5) G(x^4 - G(x^3 - G(x^2 - G(x - G(x))))) = x^5 - x^6 + x^7 - x^8 + x^9 - x^10 + x^11 - 2*x^12 + 3*x^13 - 3*x^14 + 3*x^15 - 4*x^16 + 5*x^17 +...
...
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PROG
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(PARI) {a(n)=local(A=x+x^2, G=x^(n+1)); for(i=1, n+1, A=serreverse(x-G+x^2*O(x^n)); G=x^(n+1); for(k=0, n-1, G=subst(A, x, x^(n-k+1)-G+x^2*O(x^n)))); polcoeff(A, n)}
for(n=1, 35, 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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