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A276360
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G.f. satisfies: A(x - A(x)^2) = x + 2*A(x)^2.
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19
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1, 3, 24, 276, 3858, 61092, 1056816, 19550475, 381543576, 7782820548, 164842646424, 3607654164924, 81281990795520, 1879865970374568, 44527769989124976, 1078220967132218616, 26650484274297181896, 671558570413109457264, 17234310756238557856200, 450044549619831325213920, 11949386806898017225833312, 322394088574898542428753168, 8833647058171126097908059720
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. A(x) also satisfies:
(1) A(x) = x + 3 * A( 2*x/3 + A(x)/3 )^2.
(2) A(x) = -2*x + 3 * Series_Reversion(x - A(x)^2).
(3) 2*R(x) = -x + 3 * Series_Reversion(x + 2*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/3 - R(x)/3 ) ) = x/3 + 2*R(x)/3, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*3^(n-k-1).
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EXAMPLE
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G.f.: A(x) = x + 3*x^2 + 24*x^3 + 276*x^4 + 3858*x^5 + 61092*x^6 + 1056816*x^7 + 19550475*x^8 + 381543576*x^9 + 7782820548*x^10 + 164842646424*x^11 + 3607654164924*x^12 +...
such that A(x - A(x)^2) = x + 2*A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - A(x)^2) = 2*x/3 + A(x)/3, which begins:
Series_Reversion(x - A(x)^2) = x + x^2 + 8*x^3 + 92*x^4 + 1286*x^5 + 20364*x^6 + 352272*x^7 + 6516825*x^8 + 127181192*x^9 + 2594273516*x^10 + 54947548808*x^11 +
1202551388308*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 3*x^2 - 6*x^3 - 51*x^4 - 564*x^5 - 7416*x^6 - 109764*x^7 - 1772028*x^8 - 30603930*x^9 - 558238326*x^10 - 10659285096*x^11 - 211688430204*x^12 +...
then Series_Reversion(x + 2*A(x)^2) = x/3 + 2*R(x)/3.
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MATHEMATICA
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m = 24; A[_] = 0;
Do[A[x_] = x + 3 A[2 x/3 + A[x]/3]^2 + O[x]^m // Normal, {m}];
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PROG
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(PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-F^2) - 2*F^2, #A) ); A[n]}
for(n=1, 30, 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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