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A271844 G.f. A(x) satisfies: A(x) = x + A( A(x)^2 + A(x)^4 ).

%I A271844 #9 Apr 22 2016 06:39:47
%S A271844 1,1,2,7,26,102,420,1793,7854,35106,159492,734334,3418892,16068532,
%T A271844 76135112,363283763,1744135306,8419281306,40838500796,198950342814,
%U A271844 972999755364,4775441138580,23513016382120,116111875760294,574927064750460,2853800953323468,14197997592237912,70786396399962476,353611516341840008,1769694222850151128
%N A271844 G.f. A(x) satisfies: A(x) = x + A( A(x)^2 + A(x)^4 ).
%C A271844 Compare g.f. to: C(x) = x + C( C(x)^2 - C(x)^4 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H A271844 Paul D. Hanna, <a href="/A271844/b271844.txt">Table of n, a(n) for n = 1..300</a>
%F A271844 G.f. A(x) satisfies:
%F A271844 (1) A( x - A(x^2 + x^4) ) = x.
%F A271844 (2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2+x^4)^(n+1) / (n+1)!.
%F A271844 (3) A(x) = x*exp( Sum_{n>=0} d^n/dx^n A(x^2+x^4)^(n+1)/x / (n+1)! ).
%F A271844 a(n) ~ c * d^n / n^(3/2), where d = 5.26908951612012208739853420341892... and c = 0.063610446185354820395355587671... . - _Vaclav Kotesovec_, Apr 16 2016
%e A271844 G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 102*x^6 + 420*x^7 + 1793*x^8 + 7854*x^9 + 35106*x^10 + 159492*x^11 + 734334*x^12 +...
%e A271844 where A(x) = x + A( A(x)^2 + A(x)^4 ).
%e A271844 RELATED SERIES.
%e A271844 A(x)^2 + A(x)^4 = x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 84*x^6 + 340*x^7 + 1434*x^8 + 6226*x^9 + 27632*x^10 + 124820*x^11 + 572000*x^12 +...
%e A271844 A(x^2 + x^4) = x^2 + 2*x^4 + 4*x^6 + 14*x^8 + 60*x^10 + 276*x^12 + 1320*x^14 + 6530*x^16 + 33188*x^18 + 172252*x^20 + 909016*x^22 +...
%e A271844 where the series reversion of A(x) equals x - A(x^2 + x^4).
%o A271844 (PARI) {a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1,n, A = x + subst(A,x,A^2 + A^4) ) ; polcoeff(A,n)}
%o A271844 for(n=1,30,print1(a(n),", "))
%Y A271844 Cf. A155134, A141200.
%K A271844 nonn
%O A271844 1,3
%A A271844 _Paul D. Hanna_, Apr 15 2016