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A141141 The main diagonal in the table of coefficients of iterations of G(x), where G(x) = x + x*G(G(x)) = g.f. of A030266.

%I A141141 #2 Mar 30 2012 18:37:10
%S A141141 1,2,12,114,1435,22219,406441,8559852,203792337,5409449156,
%T A141141 158350300141,5066765087000,175908765569628,6585443884172129,
%U A141141 264428161094825151,11335716352419699208,516717363793695685925,24955728581736822645816
%N A141141 The main diagonal in the table of coefficients of iterations of G(x), where G(x) = x + x*G(G(x)) = g.f. of A030266.
%e A141141 a(n) = the n-th coefficient of the n-th iteration of G(x):
%e A141141 [x] G(x) = 1, [x^2] G(G(x)) = 2, [x^3] G(G(G(x))) = 12, etc.
%e A141141 The initial iterations (n=1..7) of G(x) are:
%e A141141 n=1: x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + 531*x^7 +...
%e A141141 n=2: x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 531*x^6 + 2982*x^7 +...
%e A141141 n=3: x + 3*x^2 + 12*x^3 + 57*x^4 + 305*x^5 + 1787*x^6 + 11269*x^7 +...
%e A141141 n=4: x + 4*x^2 + 20*x^3 + 114*x^4 + 712*x^5 + 4772*x^6 + 33896*x^7 +...
%e A141141 n=5: x + 5*x^2 + 30*x^3 + 200*x^4 + 1435*x^5 + 10900*x^6 + 86799*x^7+...
%e A141141 n=6: x + 6*x^2 + 42*x^3 + 321*x^4 + 2608*x^5 + 22219*x^6 + 196910*x^7+...
%e A141141 n=7: x + 7*x^2 + 56*x^3 + 483*x^4 + 4389*x^5 + 41531*x^6 + 406441*x^7+...
%e A141141 Notice the main diagonal of the table formed from these coefficients.
%o A141141 (PARI) {a(n)=local(A=x,B); if(n<1, 0, for(i=1, n, A=serreverse(x/(1+A +x*O(x^n)))); B=x;for(i=1,n,B=subst(A,x,B+x*O(x^n)));polcoeff(B,n))}
%Y A141141 Cf. A030266, A128325, A128329.
%K A141141 nonn
%O A141141 1,2
%A A141141 _Paul D. Hanna_, Jun 10 2008