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%I #5 Sep 02 2017 04:15:44
%S 1,1,7,67,769,10009,143359,2218255,36625657,639659737,11741022235,
%T 225390779647,4508109360985,93665093491381,2016669357747667,
%U 44905700922069463,1032419000661778213,24472819932819733957,597384952530618840715,15000294032677574361955,387082666821619977435277,10256260095368150955828565,278811213889895147327704519,7770474960716476086765483619
%N G.f. A(x) satisfies: A(x - 3*x*A(x)) = x - 2*x*A(x).
%F G.f. A(x) also satisfies:
%F (1) A(x) = (1/3)*Series_Reversion( x - 3*x*A(x) ) + 2*x/3.
%F (2) A( 3*A(x) - 2*x) ) = (A(x) - x) / (3*A(x) - 2*x).
%F a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k.
%e G.f.: A(x) = x + x^2 + 7*x^3 + 67*x^4 + 769*x^5 + 10009*x^6 + 143359*x^7 + 2218255*x^8 + 36625657*x^9 + 639659737*x^10 + 11741022235*x^11 + 225390779647*x^12 +...
%e such that A(x - 3*x*A(x)) = x - 2*x*A(x).
%e RELATED SERIES.
%e A(x - 3*x*A(x)) = x - 2*x^2 - 2*x^3 - 14*x^4 - 134*x^5 - 1538*x^6 - 20018*x^7 +...
%e which equals x - 2*x*A(x).
%e Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 21*x^3 + 201*x^4 + 2307*x^5 + 30027*x^6 + 430077*x^7 + 6654765*x^8 +...
%e which equals 3*A(x) - 2*x.
%e A( 3*A(x) - 2*x ) = x + 4*x^2 + 34*x^3 + 382*x^4 + 5038*x^5 + 74134*x^6 + 1184650*x^7 + 20224990*x^8 + 364994554*x^9 + 6911857450*x^10 + 136622440786*x^11 + 2807805653098*x^12 +...
%e which equals (A(x) - x) / (3*A(x) - 2*x).
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = (1/3)*serreverse( x - 3*x*A +x*O(x^n) ) + 2*x/3 ); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A291820, A291813, A291815, A291816, A291817, A291818, A291819, A276358, A291743, A291744.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Sep 02 2017
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