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%I #11 Oct 15 2019 09:20:07
%S 1,-1,3,-21,180,-1728,17766,-191322,2130381,-24328755,283378095,
%T -3353616513,40209584220,-487400777388,5963013401400,-73536394960872,
%U 913148848213740,-11408099402698488,143288658075653370,-1808347110471143550,22919638640740373070,-291612557490911415990,3723227753617063771380,-47688275868038913956940,612581097400862637697650,-7889900533870136800820958
%N G.f. A(x) satisfies: x = 1 + 2*A(x) - 6*A(x)^2 + 3*A(x)^3.
%H Paul D. Hanna, <a href="/A295541/b295541.txt">Table of n, a(n) for n = 0..500</a>
%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%F G.f. A(x) satisfies:
%F (1) A(x) = 1 - Series_Reversion( x + 3*x^2 - 3*x^3 ).
%F (2) 1/A(x) = 1 + Series_Reversion( x*(1 + 5*x + x^2)/(1+x)^3 ).
%F (3) 1 - x = A( x + 3*x^2 - 3*x^3 ).
%F (4) 1/(1+x) = A ( x*(1 + 5*x + x^2)/(1+x)^3 ).
%F (5) x = (1 - A(x)) * (1 + 3*A(x) - 3*A(x)^2).
%e G.f.: A(x) = 1 - x + 3*x^2 - 21*x^3 + 180*x^4 - 1728*x^5 + 17766*x^6 - 191322*x^7 + 2130381*x^8 - 24328755*x^9 + 283378095*x^10 +...
%e such that x = (1 - A(x)) * (1 + 3*A(x) - 3*A(x)^2).
%e RELATED SERIES.
%e x/(1 - A(x)) = 1 + 3*x - 12*x^2 + 81*x^3 - 693*x^4 + 6642*x^5 - 68229*x^6 + 734346*x^7 - 8173791*x^8 + 93317103*x^9 - 1086705477*x^10 +...
%e which equals 1 + 3*A(x) - 3*A(x)^2.
%e 1/A(x) = 1 + x - 2*x^2 + 16*x^3 - 137*x^4 + 1321*x^5 - 13610*x^6 + 146770*x^7 - 1635878*x^8 + 18694960*x^9 - 217876232*x^10 +...
%e which equals 1 - Series_Reversion( x*(1 + 5*x + x^2)/(1+x)^3 ).
%e A(x)^2 = 1 - 2*x + 7*x^2 - 48*x^3 + 411*x^4 - 3942*x^5 + 40509*x^6 - 436104*x^7 + 4854978*x^8 - 55434456*x^9 + 645613254*x^10 +...
%e A(x)^3 = 1 - 3*x + 12*x^2 - 82*x^3 + 702*x^4 - 6732*x^5 + 69174*x^6 - 744660*x^7 + 8289702*x^8 - 94649742*x^9 + 1102307778*x^10 +...
%e where x = 1 + 2*A(x) - 6*A(x)^2 + 3*A(x)^3.
%o (PARI) {a(n) = my(A=1); A = 1 - serreverse( subst( (1-x)*(1 + 3*x - 3*x^2), x, 1-x) +x^2*O(x^n) ); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=1); A = 1/(1 + serreverse( x*(1 + 5*x + x^2)/(1+x)^3 +x^2*O(x^n)) ); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%K sign
%O 0,3
%A _Paul D. Hanna_, Nov 24 2017
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