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%I #53 Aug 22 2022 04:50:31
%S 0,1,2,2,0,8,-56,32,10176,-215808,-78784,150990912,-3405688576,
%T -139041794560,10385778676736,130003936220160,-43016304236761088,
%U 526545841919713280,266085261164348628992,-12347306589339686547456
%N Coefficients of the formal power series of a half-iterate of exp(x)-1 (rescaled).
%C Consider the formal power series for the real half-iterate of exp(x)-1 = Sum_{k>=0} c_k*x^k with c_1 = +1 then a(k) = c_k*k!*4^{k-1} and all a(k) seem to be integers.
%C For the general technique of finding the half-iterate of power series, see for instance the Comtet reference.
%D Comtet, L; Advanced Combinatorics (1974 edition), D. Reidel Publishing Company, Dordrecht - Holland, pp. 147-148.
%H Gottfried Helms, <a href="https://go.helms-net.de/math/tetdocs/CoefficientsForUTetration.htm">Coefficients for fractional iterates exp(x)-1</a>
%H Dmitry Kruchinin and Vladimir Kruchinin, <a href="https://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 G.f. f(x) where f(f(x)) = exp(x)-1 with f'(0)=1.
%F T(n,m) = if n=m then 1 else (stirling2(n,m)*m!/n!-sum(i=m+1..n-1, T(n,i)*T(i,m)))/2; a(n) = 4^(n-1)*n!*T(n,1). - _Vladimir Kruchinin_, Nov 09 2011
%F E.g.f. A(x), satisfies A(A(x))=(exp(4*x)-1)/4, T(n,m)=1/2*(4^(n-m)*stirling2(n,m)-sum(i=m+1..n-1, T(n,i)*T(i,m))), T(n,n)=1, a(n)=T(n,1), a(0)=0. - _Dmitry Kruchinin_, Dec 04 2012
%F a(n) = A052122(n) * 2^(2*n - 2 - A052123(n)). - _Andrey Zabolotskiy_, Aug 22 2022
%e f(x) = x + 1/4*x^2 + 1/48*x^3 + 1/3840*x^5 - 7/92160*x^6 + 1/645120*x^7 + O(x^8)
%e so c_3 = 1/48
%e and a(3) = c_3 * 4^2*3! = 16*6/48 = 2
%t max = 19; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; coes = CoefficientList[ Series[f[f[x]] - Exp[x] - 1, {x, 0, max}], x]; sol = Solve[Thread[coes == 0] // Rest] // First; Table[c[n]*4^(n-1)*n!, {n, 0, max}] /. sol (* _Jean-François Alcover_, Feb 11 2013 *)
%o (PARI) {a(n)=local(A=x+x^2,B=x);for(i=1,n,B=serreverse(A+x*O(x^n));A=(A+exp(B)-1)/2);4^(n-1)*n!*polcoeff(A,n)} \\ _Paul D. Hanna_
%o (PARI)
%o {trisqrt(m) = local(tmp, rs=rows(m), cs=cols(m), c);
%o \\ computes sqrt of lower triangular matrix with unit-diagonal
%o tmp=matid(#m);
%o for(d=1,rs-1,
%o for(r=d+1,rs,
%o c=r-d;
%o tmp[r,c]=(m[r,c]-sum(k=c+1,r-1,tmp[r,k]*tmp[k,c]))
%o /(tmp[c,c]+tmp[r,r])
%o );
%o );
%o return(tmp);}
%o ff = exp(x)-1
%o Mff = matrix(6,6,r,c,polcoeff(ff^(c-1),(r-1))) \\ create Bell-matrix for ff
%o Mf = trisqrt ( Mff ) \\ = Mff^(1/2) is Bellmatrix for f
%o f = Ser(Mf[,2]) \\ coefficients of power series for half-iterate of exp(x)-1 from second column in Mf
%Y Cf. A052122, A052123.
%K sign
%O 0,3
%A _Gottfried Helms_, Feb 13 2011
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