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(2) x^2 = A(A(x))^2 - 8*A(A(x))^3, where 2*A(A(x/2)) is the g.f. of A078531.

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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) x = A( sqrt(A(x)^2 - 8*A(x)^3) ).

(2) x^2 = A(A(x))^2 - 8*A(A(x))^3, where A(A(x)) is the g.f. of A078531.

(3) [x^(n+1)] A(A(x)) = 8^n * binomial((3*n-1)/2, n)/(n+1) = 2^n*A078531(n) for n >= 0.

also

Also,

Ai(x)/A(x) = sqrt(1 - 8*A(x)) = 1 - 4*x - 16+ 4*x^2 - 128+ 40*x^3 - 1328+ 512*x^4 - 15840+ 7392*x^5 - 205856+ 114688*x^6 - 2831616+ 1867008*x^7 - 40515392+ 31457280*x^8 - + 543921664*x^9 + ... + 2^n*A078531(n)*x^(n+1) + ...

which satisfies A(A(x))^2 - 8*A(A(x))^3 = x^2, where

A(A(x))^2 = x^2 + 8*x^3 + 96*x^4 + 1344*x^5 + 20480*x^6 + 329472*x^7 + ...

A(A(x))^3 = x^3 + 12*x^4 + 168*x^5 + 2560*x^6 + 41184*x^7 + 688128*x^8 + ...

Cf. A078531, A273925.

allocated for Paul D. Hanna

Expansion of g.f. A(x) satisfying A( sqrt(A(x)^2 - 8*A(x)^3) ) = x.

1, 2, 16, 172, 2120, 28264, 396192, 5746480, 85394656, 1291778368, 19805198784, 306834276416, 4793670528640, 75415927948416, 1193652980090880, 18994846756882176, 303766882134726144, 4880209392051146752, 78739290124904116224, 1275444751485628848128, 20735204112205333970944

1,2

G.f.: A(x) = x + 2*x^2 + 16*x^3 + 172*x^4 + 2120*x^5 + 28264*x^6 + 396192*x^7 + 5746480*x^8 + 85394656*x^9 + 1291778368*x^10 + ...

where A( sqrt(A(x)^2 - 8*A(x)^3) ) = x.

RELATED SERIES.

A(x)^2 = x^2 + 4*x^3 + 36*x^4 + 408*x^5 + 5184*x^6 + 70512*x^7 + 1002864*x^8 + 14711456*x^9 + 220670592*x^10 + ...

A(x)^3 = x^3 + 6*x^4 + 60*x^5 + 716*x^6 + 9384*x^7 + 130344*x^8 + 1882576*x^9 + 27950736*x^10 + ...

Let Ai(x) be the series reversion of A(x), then

Ai(x)^2 = A(x)^2 - 8*A(x)^3 = x^2 - 4*x^3 - 12*x^4 - 72*x^5 - 544*x^6 - 4560*x^7 - 39888*x^8 - 349152*x^9 - 2935296*x^10 - ...

and

Ai(x) = sqrt(A(x)^2 - 8*A(x)^3) = x - 2*x^2 - 8*x^3 - 52*x^4 - 408*x^5 - 3512*x^6 - 31584*x^7 - 287056*x^8 - 2560288*x^9 - ...

also

Ai(x)/A(x) = sqrt(1 - 8*A(x)) = 1 - 4*x - 16*x^2 - 128*x^3 - 1328*x^4 - 15840*x^5 - 205856*x^6 - 2831616*x^7 - 40515392*x^8 - ...

(PARI) {a(n) = my(A=1, V=[1]); for(i=1, n, V = concat(V, 0); A = x*Ser(V);

V[#V] = polcoeff( x - subst(A, x, sqrt(A^2 - 8*A^3)), #V)/2 ); V[n]}

for(n=1, 30, print1(a(n), ", "))

Cf. A273925.

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nonn

Paul D. Hanna, Dec 29 2023

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allocated for Paul D. Hanna

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