A119818 -id:A119818 - cp's OEIS frontend

A119817 Integer a(n) produces the least nonnegative integer coefficient of x^n in the n-th iteration of g.f. A(x).

1, 1, -2, 8, -40, 210, -1032, 4074, -9084, -1485, -139344, -1178057, 97107644, 533077818, -43465435335, -997494915376, 35039558716800, 1885975569825115, -36684866143759995, -4946226556607087316, 24828007395162323458, 18213320246807011794109

Offset: 1

Paul D. Hanna, May 31 2006

sign

			The successive iterations of g.f. A(x) begin:
A(x) = (1)x + x^2 - 2x^3 + 8x^4 - 40x^5 + 210x^6 - 1032x^7 + 4074x^8+..
A(A(x)) = x + (2)x^2 - 2x^3 + 7x^4 - 30x^5 + 118x^6 -268x^7 -1430x^8+..
A(A(A(x))) = x + 3x^2 + (0)x^3 + 3x^4 -12x^5 +18x^6 +240x^7 -3119x^8+..
A(A(A(A(x)))) = x + 4x^2 + 4x^3 + (2)x^4 - 4x^5 - 18x^6 + 276x^7+...
A(A(A(A(A(x))))) = x + 5x^2 + 10x^3 + 10x^4 +(0)x^5 -20*x^6 +128*x^7+..
A(A(A(A(A(A(x)))))) = x + 6x^2 + 18x^3 +33x^4 +30x^5 +(0)x^6 -24x^7+..
Coefficients [x^n] of n-th iteration of A(x) forms 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,...].

Cf. A119818, A119815, A112317.

{a(n)=local(F=x+x^2+sum(k=3,n-1,a(k)*x^k),G=x+x*O(x^n)); if(n<1,0,if(n<=2,1, for(k=1,n,G=subst(F,x,G)); return((n-1-polcoeff(G,n,x)) )))}

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.

Original entry on oeis.org

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

Paul D. Hanna, May 31 2006

nonn

For prime p, a(p) = p; for all n>=1, 0 < a(n) <=n.

			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 +...

Paul D. Hanna, Table of n, a(n) for n = 1..185

Cf. A119815, A119818.

{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]))}

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.

cp's OEIS Frontend

A119817 Integer a(n) produces the least nonnegative integer coefficient of x^n in the n-th iteration of g.f. A(x).

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