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.
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, 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, 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
Offset: 1
Keywords
Examples
Let F(x) = g.f. of A119817 = [1,1,-2,8,-40,210,-1032,4074,-9084,...], then the coefficient of x^n in the n-th iteration of F(x) forms [1,2,0,2,0,0,0,0,0,0,0,10,...], as illustrated by: F(x) = (1)x + x^2 - 2x^3 + 8x^4 - 40x^5 + 210x^6 - 1032x^7 + 4074x^8+.. F(F(x)) = x + (2)x^2 - 2x^3 + 7x^4 - 30x^5 + 118x^6 -268x^7 -1430x^8+.. F(F(F(x))) = x + 3x^2 + (0)x^3 + 3x^4 -12x^5 +18x^6 +240x^7 -3119x^8+.. F(F(F(F(x)))) = x + 4x^2 + 4x^3 + (2)x^4 - 4x^5 - 18x^6 + 276x^7+... F(F(F(F(F(x))))) = x + 5x^2 + 10x^3 + 10x^4 +(0)x^5 -20*x^6 +128*x^7+.. F(F(F(F(F(F(x)))))) = x + 6x^2 + 18x^3 +33x^4 +30x^5 +(0)x^6 -24x^7+..
Programs
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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]))}
Formula
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.
Comments