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%I #6 Dec 03 2017 20:01:33
%S 1,3,6,24,96,396,1728,7839,36438,172680,831624,4058202,20021268,
%T 99697188,500429016,2529375300,12862429920,65760468840,337817930184,
%U 1742850773154,9026374329108,46912014922392,244588357460448,1278937818954306,6705339839722404,35241796466506908,185643541655678184,979972436105339856,5183169679909147200,27464173024052341200
%N G.f. A(x) satisfies: A(x - A(x^2)) = x + 2*A(x^2).
%H Paul D. Hanna, <a href="/A295761/b295761.txt">Table of n, a(n) for n = 1..1030</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = x + 3 * A( (2*x + A(x))^2/9 ).
%F (2) A(x) = -2*x + 3 * Series_Reversion( x - A(x^2) ).
%F (3) x = A( -x/2 + 3/2 * Series_Reversion( x + 2*A(x^2) ) ).
%F (4) A(x - A(x^2)) = x + 2*A(x^2).
%e G.f.: A(x) = x + 3*x^2 + 6*x^3 + 24*x^4 + 96*x^5 + 396*x^6 + 1728*x^7 + 7839*x^8 + 36438*x^9 + 172680*x^10 + 831624*x^11 + 4058202*x^12 +...
%e such that A(x - A(x^2)) = x + 2*A(x^2).
%e RELATED SERIES.
%e A(x - A(x^2)) = x + 2*x^2 + 6*x^4 + 12*x^6 + 48*x^8 + 192*x^10 + 792*x^12 + 3456*x^14 + 15678*x^16 + 72876*x^18 + 345360*x^20 + 1663248*x^22 + 8116404*x^24 + 40042536*x^26 + 199394376*x^28 + 1000858032*x^30 + 5058750600*x^32 +...
%e which equals x + 2*A(x^2).
%e Series_Reversion( x - A(x^2) ) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 132*x^6 + 576*x^7 + 2613*x^8 + 12146*x^9 + 57560*x^10 + 277208*x^11 + 1352734*x^12 +...
%e which equals (A(x) + 2*x)/3.
%e A( (2*x + A(x))^2/9 ) = x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 132*x^6 + 576*x^7 + 2613*x^8 + 12146*x^9 + 57560*x^10 + 277208*x^11 + 1352734*x^12 +...
%e which equals (A(x) - x)/3.
%o (PARI) {a(n) = my(A=x); for(i=1,n, A = -2*x + 3*serreverse(x - subst(A,x,x^2) +x^2*O(x^n)) ); polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A141200, A295760, A295762.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Dec 03 2017
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