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A272485
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G.f. A(x) satisfies: A(x)^3 = x * A( A(x)^2 + A(x)^3 ).
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2
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1, 1, 2, 6, 20, 72, 274, 1084, 4413, 18365, 77773, 334065, 1451929, 6373379, 28214851, 125826137, 564736106, 2549004457, 11563003842, 52688331678, 241049183598, 1106818411361, 5098961600041, 23561122837102, 109171473224810, 507136264223327, 2361332083301582, 11018704225690520, 51520180227572593, 241344269535483201, 1132543412040153639, 5323310040423904420
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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( x^3 / A(x^2 + x^3) ) = x.
a(n) ~ c * d^n / n^(3/2), where d = 4.93477240448054430929819... and c = 0.06099488383521219235235... . - Vaclav Kotesovec, May 07 2016
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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 72*x^6 + 274*x^7 + 1084*x^8 + 4413*x^9 + 18365*x^10 + 77773*x^11 + 334065*x^12 +...
where A(x)^3 = x * A( A(x)^2 + A(x)^3 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 56*x^6 + 208*x^7 + 808*x^8 + 3244*x^9 + 13354*x^10 + 56060*x^11 + 239080*x^12 + 1032908*x^13 + 4511152*x^14 + 19884256*x^15 +...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 438*x^8 + 1742*x^9 + 7116*x^10 + 29685*x^11 + 125937*x^12 + 541704*x^13 + 2357028*x^14 + 10355984*x^15 +...
A(x^2 + x^3) = x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 6*x^7 + 12*x^8 + 26*x^9 + 56*x^10 + 124*x^11 + 278*x^12 + 632*x^13 + 1454*x^14 + 3378*x^15 + 7918*x^16 + 18694*x^17 + 44427*x^18 + 106175*x^19 + 255031*x^20 + 615320*x^21 + 1490588*x^22 +...
Let B(x) be the series reversion of g.f. A(x), then
B(x) = x - x^2 - x^4 - 2*x^6 - 2*x^7 - 7*x^8 - 11*x^9 - 28*x^10 - 57*x^11 - 134*x^12 - 300*x^13 - 700*x^14 - 1627*x^15 - 3833*x^16 - 9075*x^17 - 21628*x^18 - 51818*x^19 - 124727*x^20 - 301523*x^21 - 731662*x^22 - 1781632*x^23 +...
where B(x) = x^3 / A(x^2 + x^3).
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PROG
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(PARI) /* From definition A( A(x)^2 + A(x)^3 ) = A(x)^3/x */
{a(n) = my(A=[1, 1, 0]); for(i=1, n, A = concat(A, 0); G=x*Ser(A); A[#A-1] = Vec(x*subst(G, x, G^2 + G^3)/G^3)[#A-1] ); A[n]}
for(n=1, 40, print1(a(n), ", "))
(PARI) /* Using Series Reversion of x^3/A(x^2+x^3) */
{a(n) = my(A=x); for(i=1, n, A = serreverse( x^3/subst(A, x, x^2 + x^3 +x^2*O(x^n))) ); polcoeff(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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