A119816 Least positive integers that can appear as the coefficient of x^n the n-th iteration of an integer function F(x) where F(0)=0, for n>=1; the F(x) that satisfies this minimal condition is the g.f. of A119815.
1, 2, 3, 4, 5, 4, 7, 8, 3, 9, 11, 4, 13, 11, 14, 8, 17, 4, 19, 4, 1, 4, 23, 24, 5, 17, 27, 22, 29, 16, 31, 32, 24, 4, 30, 36, 37, 4, 36, 10, 41, 18, 43, 41, 17, 27, 47, 40, 28, 29, 7, 10, 53, 9, 1, 24, 49, 4, 59, 57, 61, 35, 31, 48, 39, 16, 67, 24, 51, 9, 71, 46, 73, 4, 56, 11, 55, 62, 79
Offset: 1
Keywords
Examples
Let F(x) = g.f. of A119815 = [1,1,-1,1,1,-11,23,-20,731,-4860,...], then the coefficient of x^n in the n-th iteration of F(x) forms [1,2,3,4,5,4,7,8,3,9,11,...], as illustrated by: F(x) = (1)x + x^2 - x^3 + x^4 + x^5 - 11x^6 + 23x^7 - 20x^8 + 731x^9+.. F(F(x)) = x + (2)x^2 - 2x^4 + 6x^5 - 8x^6 - 50x^7 + 78x^8 + 1688x^9+... F(F(F(x))) = x + 3x^2 + (3)x^3 - 3x^4 - x^5 + 17x^6 - 81x^7 -370x^8+... F(F(F(F(x)))) = x + 4x^2 + 8x^3 + (4)x^4 - 12x^5 + 4x^6 + 12x^7 +... F(F(F(F(F(x))))) = x + 5x^2 + 15x^3 + 25x^4 + (5)x^5 - 55x^6 -33x^7+... F(F(F(F(F(F(x)))))) = x + 6x^2 + 24x^3 + 66x^4 + 106x^5 + (4)x^6 +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..185
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-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 A119815 causes {a(n)} to be the least positive integers.
Comments