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A119818 a(n) is the least nonnegative integer that can appear as the coefficient of x^n in the n-th iteration of any integer function that begins with the same initial n-1 terms as the g.f. of A119817 for n>1, with a(1)=1.

%I A119818 #11 Jul 22 2024 00:49:27
%S A119818 1,2,0,2,0,0,0,0,0,0,0,10,0,0,7,12,0,6,0,9,2,11,0,8,10,13,18,18,0,0,0,
%T A119818 0,1,0,4,6,0,19,22,1,0,41,0,14,4,23,0,26,21,22,14,11,0,42,10,21,38,0,
%U A119818 0,46,0,31,9,40,8,33,0,16,35,7,0,66,0,37,20,63,20,58,0,74,9,0,0,23,5,0,31,75
%N A119818 a(n) is the least nonnegative integer that can appear as the coefficient of x^n in the n-th iteration of any integer function that begins with the same initial n-1 terms as the g.f. of A119817 for n>1, with a(1)=1.
%C A119818 For prime p, a(p) = 0; for all n>=1, 0 <= a(n) < n.
%F A119818 a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(F(x)) such that F(x) = g.f. of A119817 causes {a(n)} to be the least nonnegative integers.
%e A119818 Let F(x) = g.f. of A119817 = [1,1,-2,8,-40,210,-1032,4074,-9084,...],
%e A119818 then the coefficient of x^n in the n-th iteration of F(x)
%e A119818 forms [1,2,0,2,0,0,0,0,0,0,0,10,...], as illustrated by:
%e A119818 F(x) = (1)x + x^2 - 2x^3 + 8x^4 - 40x^5 + 210x^6 - 1032x^7 + 4074x^8+..
%e A119818 F(F(x)) = x + (2)x^2 - 2x^3 + 7x^4 - 30x^5 + 118x^6 -268x^7 -1430x^8+..
%e A119818 F(F(F(x))) = x + 3x^2 + (0)x^3 + 3x^4 -12x^5 +18x^6 +240x^7 -3119x^8+..
%e A119818 F(F(F(F(x)))) = x + 4x^2 + 4x^3 + (2)x^4 - 4x^5 - 18x^6 + 276x^7+...
%e A119818 F(F(F(F(F(x))))) = x + 5x^2 + 10x^3 + 10x^4 +(0)x^5 -20*x^6 +128*x^7+..
%e A119818 F(F(F(F(F(F(x)))))) = x + 6x^2 + 18x^3 +33x^4 +30x^5 +(0)x^6 -24x^7+..
%o A119818 (PARI) {a(n)=my(A=vector(n),B,F=x+x^2,G); if(n==1||n==2,n,A[1]=1; A[2]=1; B=A; B[2]=2; for(m=3,n,G=x+x*O(x^n); for(k=1,m,G=subst(F,x,G)); B[m]=polcoeff(G,m,x); A[m]=(m-1-B[m])\m; F=F+A[m]*x^m); return(B[n]+n*A[n]))}
%Y A119818 Cf. A119817, A119816, A112317.
%K A119818 nonn
%O A119818 1,2
%A A119818 _Paul D. Hanna_, May 31 2006