A052104 Numerators of coefficients of the formal power series a(x) such that a(a(x)) = exp(x) - 1.

0, 1, 1, 1, 0, 1, -7, 1, 53, -281, -1231, 87379, -13303471, -54313201, 10142361989, 2821265977, -10502027401553, 1836446156249, 2952828271088741, -1004826382596003137, -7006246797736924249, 14607119841651449406947, 1868869263315549659372569

Offset: 0

Author

N. J. A. Sloane, Jan 22 2000

Keywords

• sign
• nice
• easy
• frac

Examples

			a(x) = x + x^2/4 + x^3/48 + x^5/3840 - 7*x^6/92160 + x^7/645120 + ...
		

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.52c.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..120
• I. N. Baker, Zusammensetzungen ganzer Funktionen Math. Z. 69 (1) (1958) 121-163.
• Dmitry Kruchinin and Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.
• Mathoverflow, f(f(x))=exp(x)-1 and...

Crossrefs

Cf. A052105, A052122, A052123.

Programs

Maple

T:= proc(n, m) T(n, m):= `if`(n=m, 1, (Stirling2(n, m)*m!/n!-
       add(T(n,i)*T(i,m), i=m+1..n-1))/2)
    end:
a:= n-> numer(T(n, 1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 11 2013

Mathematica

T[n_, n_] = 1; T[n_, m_] := T[n, m] = (StirlingS2[n, m]*m!/n! - Sum[T[n, i]*T[i, m], {i, m+1, n-1}])/2; Table[T[n, 1] // Numerator, {n, 0, 30}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)

Sage

@CachedFunction
def T(n,k):
    if (k==n): return 1
    else: return ( (factorial(k)/factorial(n))*stirling_number2(n,k) - sum(T(n,j)*T(j,k) for j in (k+1..n-1)) )/2
[numerator(T(n,1)) for n in (0..30)] # G. C. Greubel, Apr 15 2021

Formula

a(n) = numerator(T(n,1)) where T(n, m) = if n=m then 1, otherwise ( StirlingS2(n, m)*m!/n! - Sum_{i=m+1..n-1} T(n, i)*T(i, m) )/2. - Vladimir Kruchinin, Nov 08 2011