The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A052104

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A052104

Numerators of coefficients of the formal power series a(x) such that a(a(x)) = exp(x) - 1.
(history; published version)

#55 by Peter Luschny at Mon Nov 29 07:48:06 EST 2021

STATUS

reviewed

approved

#54 by Joerg Arndt at Mon Nov 29 05:56:52 EST 2021

STATUS

proposed

reviewed

#53 by Andrey Zabolotskiy at Mon Nov 29 05:54:15 EST 2021

STATUS

editing

proposed

Discussion

Mon Nov 29

05:56

Joerg Arndt: thanks.

#52 by Andrey Zabolotskiy at Sat Nov 27 03:17:35 EST 2021

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 65.5252c.

Discussion

Sat Nov 27

05:09

Joerg Arndt: This is one solution a(x) of a(a(x)) = exp(x) - 1. But is it unique?

Mon Nov 29

05:36

Andrey Zabolotskiy: This solution is unique as a formal power series. However, it converges nowhere (see MathOverflow), hence my edit. On the other hand, they say at MathOverflow that the unique solution analytic for x>0 and for x<0 (but not in the vicinity of 0) exists and has this power series as an asymptotic expansion.

#51 by Andrey Zabolotskiy at Sat Nov 27 03:07:20 EST 2021

NAME

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

STATUS

approved

editing

#50 by N. J. A. Sloane at Thu Apr 15 23:46:58 EDT 2021

STATUS

proposed

approved

#49 by Jon E. Schoenfield at Thu Apr 15 22:07:33 EDT 2021

STATUS

editing

proposed

#48 by Jon E. Schoenfield at Thu Apr 15 22:07:31 EDT 2021

FORMULA

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

STATUS

proposed

editing

#47 by Michel Marcus at Thu Apr 15 04:46:46 EDT 2021

STATUS

editing

proposed

#46 by Michel Marcus at Thu Apr 15 04:46:42 EDT 2021

LINKS

I. N. Baker, <a href="http://dx.doi.org/10.1007/BF01187396">Zusammensetzungen ganzer Funktionen</a> Math. Z. 69 (1) (1958) 121-163.

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.

STATUS

proposed

editing