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A291813 G.f. A(x) satisfies: A(x - 2*x*A(x)) = x - x*A(x).

%I A291813 #8 Sep 02 2017 04:15:21
%S A291813 1,1,5,35,297,2873,30657,353727,4355497,56709337,775575269,
%T A291813 11085971235,164979882033,2548461481105,40762085472929,
%U A291813 673751263927071,11489101983573105,201838769635965969,3648620371959258149,67795012307507004291,1293607920940368319641,25326486746707799668105,508368313083167614599201,10454499119633293760277151,220120546753823908307191769,4742197866143368618862457641
%N A291813 G.f. A(x) satisfies: A(x - 2*x*A(x)) = x - x*A(x).
%F A291813 G.f. A(x) also satisfies:
%F A291813 (1) A(x) = (1/2)*Series_Reversion( x - 2*x*A(x) ) + x/2.
%F A291813 (2) A( 2*A(x) - x ) = (A(x) - x) / (2*A(x) - x).
%F A291813 a(n) = Sum_{k=0..n-1} A291820(n, k) * 2^k.
%e A291813 G.f.: A(x) = x + x^2 + 5*x^3 + 35*x^4 + 297*x^5 + 2873*x^6 + 30657*x^7 + 353727*x^8 + 4355497*x^9 + 56709337*x^10 + 775575269*x^11 + 11085971235*x^12 +...
%e A291813 such that  A(x - 2*x*A(x)) = x - x*A(x).
%e A291813 RELATED SERIES.
%e A291813 A(x - 2*x*A(x)) = x - x^2 - x^3 - 5*x^4 - 35*x^5 - 297*x^6 - 2873*x^7 - 30657*x^8 +...
%e A291813 which equals x - x*A(x).
%e A291813 Series_Reversion( x - 2*x*A(x) ) = x + 2*x^2 + 10*x^3 + 70*x^4 + 594*x^5 + 5746*x^6 + 61314*x^7 + 707454*x^8 + 8710994*x^9 + 113418674*x^10 +...
%e A291813 which equals 2*A(x) - x.
%e A291813 A( 2*A(x) - x ) = x + 3*x^2 + 19*x^3 + 159*x^4 + 1561*x^5 + 17087*x^6 + 202975*x^7 + 2574391*x^8 + 34495545*x^9 + 484770627*x^10 + 7107406323*x^11 + 108289787415*x^12 + 1709478736593*x^13 + 27894511442079*x^14 +...
%e A291813 which equals (A(x) - x) / (2*A(x) - x).
%o A291813 (PARI) {a(n) = my(A=x); for(i=1, n, A = (1/2)*serreverse( x - 2*x*A +x*O(x^n) ) + x/2 ); polcoeff(A, n)}
%o A291813 for(n=1, 30, print1(a(n), ", "))
%Y A291813 Cf. A291820, A291814, A291815, A291816, A291817, A291818, A291819, A276358, A291743, A291744.
%K A291813 nonn
%O A291813 1,3
%A A291813 _Paul D. Hanna_, Sep 01 2017