A381666 The generating function A(x) satisfies the functional equation: A(x)+x = x*A(A(x)).

0, -1, 1, 0, -2, 1, 10, -13, -70, 163, 585, -2162, -5361, 30588, 49870, -459125, -411370, 7257651, 1513653, -119997558, 56857538, 2062729507, -2444340720, -36662245639, 71849171621, 670108236318, -1904023701457, -12520858710212, 48731008916451, 237412587011506, -1237341547854760

Offset: 0

Author

Thomas Scheuerle, Mar 03 2025

Keywords

• sign
• eigen

Comments

Shifts left under COMPOSE transform with itself.

Examples

			G.f.: A(x) = -x + x^2 - 2*x^4 + x^5 + 10*x^6 + ...
A(A(x)) = x - 2*x^3 + x^4 + 10*x^5 - 13*x^6 + ...
		

Links

• N. J. A. Sloane, Transforms.

Crossrefs

Cf. A030266 ( A(x)-x = x*A(A(x)) ).

Cf. A347080 ( A(x)-x = x*A(A(-x)) ).

Cf. A381669, A381670.

Programs

PARI

a(n) = { my(A=-1+x); for(i=0, n, A=-1+x*A*subst(A, x, x*A+x*O(x^n))); if(n==0,0,polcoeff(A, n-1))}

Formula

Let a(n) = b(n, 1), with b(1, m) = -1 and b(0, m) = 0, then

b(n, m) = Sum_{k=0..n-1} (-1)^(n-1)*m*binomial(n + m - 1, k)/(n + m - 1) * b(n - k, k).