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A194972 a(n) equals the coefficient of x^n in the (2*n-1)-th iteration of x+x^2 for n>=1.

%I A194972 #7 Mar 30 2012 18:37:28
%S A194972 1,3,20,231,3864,85140,2332616,76485227,2921536088,127421864328,
%T A194972 6248486040840,340321635330534,20383240346962440,1331538898625750100,
%U A194972 94216429100347571448,7178425650032302557691,585936966156456139931584,51011156414845408925712816
%N A194972 a(n) equals the coefficient of x^n in the (2*n-1)-th iteration of x+x^2 for n>=1.
%e A194972 The coefficients of x^k, k>=1, in the odd iterations of x+x^2 begin:
%e A194972 n=1: [(1), 1];
%e A194972 n=3: [1,(3), 6, 9, 10, 8, 4, 1];
%e A194972 n=5: [1, 5,(20), 70, 220, 630, 1656, 4014, 8994, 18654, ...];
%e A194972 n=7: [1, 7, 42,(231), 1190, 5810, 27076, 121023, 520626, ...];
%e A194972 n=9: [1, 9, 72, 540,(3864), 26628, 177744, 1153740, 7303164, ...];
%e A194972 n=11:[1, 11, 110, 1045, 9570,(85140), 739332, 6286797, ...];
%e A194972 n=13:[1, 13, 156, 1794, 20020, 218218,(2332616), 24519066, ...];
%e A194972 n=15:[1, 15, 210, 2835, 37310, 481390, 6110468,(76485227), ...]; ...
%e A194972 coefficients in parenthesis form the initial terms of this sequence.
%o A194972 (PARI) {a(n)=local(A=x,G=x+x^2); for(i=1,2*n-1, A=subst(G, x, A+x*O(x^n))); polcoeff(A,n)}
%Y A194972 Cf. A194971, A122888, A112317.
%K A194972 nonn
%O A194972 1,2
%A A194972 _Paul D. Hanna_, Sep 06 2011