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A276362
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G.f. satisfies: A(x - A(x)^2) = x + 3*A(x)^2.
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19
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1, 4, 40, 564, 9592, 184008, 3844624, 85700980, 2011283640, 49248127800, 1250064156912, 32736194249256, 881252194701616, 24317581366876880, 686300288661644960, 19774058901706750100, 580795172081872246232, 17368587281321383296184, 528294942152813411073968, 16329939570298980826852824, 512590568042639978453793744, 16329084800479729420462546352, 527621994750854274463428080608
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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 + 4 * A( 3*x/4 + A(x)/4 )^2.
(2) A(x) = -3*x + 4 * Series_Reversion(x - A(x)^2).
(3) 3*R(x) = -x + 4 * Series_Reversion(x + 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/3 - R(x)/3 ) ) = x/4 + 3*R(x)/4, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*4^(n-k-1).
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EXAMPLE
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G.f.: A(x) = x + 4*x^2 + 40*x^3 + 564*x^4 + 9592*x^5 + 184008*x^6 + 3844624*x^7 + 85700980*x^8 + 2011283640*x^9 + 49248127800*x^10 + 1250064156912*x^11 + 32736194249256*x^12 +...
such that A(x - A(x)^2) = x + 3*A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - A(x)^2) = 3*x/4 + A(x)/4, which begins:
Series_Reversion(x - A(x)^2) = x + x^2 + 10*x^3 + 141*x^4 + 2398*x^5 + 46002*x^6 + 961156*x^7 + 21425245*x^8 + 502820910*x^9 + 12312031950*x^10 + 312516039228*x^11 + 8184048562314*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 4*x^2 - 8*x^3 - 84*x^4 - 1112*x^5 - 17352*x^6 - 303824*x^7 - 5791060*x^8 - 117898648*x^9 - 2531645240*x^10 - 56835852080*x^11 - 1325547044072*x^12 +...
then Series_Reversion(x + 3*A(x)^2) = x/4 + 3*R(x)/4.
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MATHEMATICA
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m = 24; A[_] = 0;
Do[A[x_] = x + 4 A[3x/4 + A[x]/4]^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) - 3*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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