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EXAMPLE
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The coefficients of the 2^(n-1)-th iterations of the g.f. begin:
(1),1,2,20,804,108304,49833296,87606851264,641794234287360,...
1,(2),6,51,1750,222706,100558052,175666197420,1284466715882828,...
1,4,(20),170,4340,474238,204872756,353171251288,2572462315656538,...
1,8,72,(804),15560,1128036,426923128,713954691088,5159170997828364,...
1,16,272,5000,(108304),4271464,962562608,1461234395040,...
1,32,1056,35856,1266720,(49833296),3774562656,3128786120000,...
1,64,4160,273440,18169920,1226585248,(87606851264),12455033590400,...
1,128,16512,2140224,278454400,36359377216,4771446963584,(641794234287360),...
in which the main diagonal forms this sequence shift left.
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PROGRAM
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(PARI) {a(n)=local(F=x+x^2+sum(m=3, n-1, a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0, if(n<=2, 1, for(i=1, n-1, G=subst(F, x, G); F=G); return(polcoeff(G, n-1, x))))}
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