Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #29 Aug 15 2021 15:08:59
%S 1,2,10,80,872,11928,195072,3702080,80065792,1950808000,53016791360,
%T 1587229842688,51619520360960,1808576831681536,68562454975587328,
%U 2830905156661645312,124395772159835529216,5504660984739184156672,250011277837808237105152,14799530615476409472303104
%N E.g.f. A(x) satisfies A(A(x)) = (1/4)*log(1/(1-4*x)).
%C a(23) = -4050933314339181211663673622528 is the first negative term. - _Vladimir Reshetnikov_, Aug 15 2021
%D Comtet, L; Advanced Combinatorics (1974 edition), D. Reidel Publishing Company, Dordrecht - Holland, pp. 147-148.
%H Vladimir Reshetnikov, <a href="/A220112/b220112.txt">Table of n, a(n) for n = 1..281</a>
%H Gottfried Helms, <a href="http://go.helms-net.de/math/tetdocs/CoefficientsForUTetration.htm">Coefficients for fractional iterates exp(x)-1</a>
%H 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
%F a(n) = T(n,1), T(n,m) = (1/2)*(4^(n-m)*(-1)^(n-m)*Stirling1(n,m) - Sum_{i=m+1..n-1} T(n,i)*T(i,m)), T(n,n)=1.
%p A := proc(n, m) option remember; if n = m then 1 else
%p 1/2*(4^(n-m)*(-1)^(n-m)*Stirling1(n,m) - add(A(n,k)*A(k,m), k =m+1..n-1)) fi end: a := n -> A(n,1): seq(a(n), n = 1..23); # _Peter Luschny_, Aug 15 2021
%t t[n_, m_] := t[n, m] = 1/2*(4^(n - m)*(-1)^(n - m)*StirlingS1[n, m] - Sum[t[n, i]*t[i, m], {i, m+1, n-1}]); t[n_, n_] = 1; Table[t[n, 1], {n, 1, 20}] (* _Jean-François Alcover_, Feb 22 2013 *)
%o (Maxima)
%o T(n,m):=if n=m then 1 else 1/2*(4^(n-m)*(-1)^(n-m)*stirling1(n,m)-sum(T(n,i)*T(i,m),i,m+1,n-1));
%o makelist((T(n,1)),n,1,10);
%Y Cf. A052122, A052123, A184011.
%K sign
%O 1,2
%A _Dmitry Kruchinin_, Dec 05 2012
%E More terms from _Vladimir Reshetnikov_, Aug 15 2021