cp's OEIS Frontend

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

%I A052104 #58 Apr 19 2025 04:29:39
%S A052104 0,1,1,1,0,1,-7,1,53,-281,-1231,87379,-13303471,-54313201,10142361989,
%T A052104 2821265977,-10502027401553,1836446156249,2952828271088741,
%U A052104 -1004826382596003137,-7006246797736924249,14607119841651449406947,1868869263315549659372569
%N A052104 Numerators of coefficients of the formal power series a(x) such that a(a(x)) = exp(x) - 1.
%D A052104 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.52c.
%H A052104 Alois P. Heinz, <a href="/A052104/b052104.txt">Table of n, a(n) for n = 0..120</a>
%H A052104 I. N. Baker, <a href="http://dx.doi.org/10.1007/BF01187396">Zusammensetzungen ganzer Funktionen</a> Math. Z. 69 (1) (1958) 121-163.
%H A052104 Dmitry Kruchinin and Vladimir Kruchinin, <a href="http://arxiv.org/abs/1302.1986">Method for solving an iterative functional equation A^{2^n}(x)=F(x)</a>, arXiv:1302.1986 [math.CO], 2013.
%H A052104 Mathoverflow, <a href="http://mathoverflow.net/questions/4347/ffxexpx-1-and-other-functions-just-in-the-middle-between-linear-and-expo">f(f(x))=exp(x)-1 and...</a>
%F A052104 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
%e A052104 a(x) = x + x^2/4 + x^3/48 + x^5/3840 - 7*x^6/92160 + x^7/645120 + ...
%p A052104 T:= proc(n, m) T(n, m):= `if`(n=m, 1, (Stirling2(n, m)*m!/n!-
%p A052104        add(T(n,i)*T(i,m), i=m+1..n-1))/2)
%p A052104     end:
%p A052104 a:= n-> numer(T(n, 1)):
%p A052104 seq(a(n), n=0..30);  # _Alois P. Heinz_, Feb 11 2013
%t A052104 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_ *)
%o A052104 (Sage)
%o A052104 @CachedFunction
%o A052104 def T(n,k):
%o A052104     if (k==n): return 1
%o A052104     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
%o A052104 [numerator(T(n,1)) for n in (0..30)] # _G. C. Greubel_, Apr 15 2021
%Y A052104 Cf. A052105, A052122, A052123.
%K A052104 sign,nice,easy,frac
%O A052104 0,7
%A A052104 _N. J. A. Sloane_, Jan 22 2000