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 A184011 Coefficients of the formal power series of a half-iterate of exp(x)-1 (rescaled). 6
 0, 1, 2, 2, 0, 8, -56, 32, 10176, -215808, -78784, 150990912, -3405688576, -139041794560, 10385778676736, 130003936220160, -43016304236761088, 526545841919713280, 266085261164348628992, -12347306589339686547456 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. For the general technique of finding the half-iterate of power series, see for instance the Comtet reference. REFERENCES Comtet, L; Advanced Combinatorics (1974 edition), D. Reidel Publishing Company, Dordrecht - Holland, pp. 147-148. LINKS Table of n, a(n) for n=0..19. Gottfried Helms, Coefficients for fractional iterates exp(x)-1 Dmitry Kruchinin and Vladimir Kruchinin, Method for solving an iterative functional equation \$A^{2^n}(x)=F(x)\$, arXiv:1302.1986 [math.CO], 2013. FORMULA G.f. f(x) where f(f(x)) = exp(x)-1 with f'(0)=1. 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 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 a(n) = A052122(n) * 2^(2*n - 2 - A052123(n)). - Andrey Zabolotskiy, Aug 22 2022 EXAMPLE 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) so c_3 = 1/48 and a(3) = c_3 * 4^2*3! = 16*6/48 = 2 MATHEMATICA 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 *) PROG (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 (PARI) {trisqrt(m) = local(tmp, rs=rows(m), cs=cols(m), c); \\ computes sqrt of lower triangular matrix with unit-diagonal tmp=matid(#m); for(d=1, rs-1, for(r=d+1, rs, c=r-d; tmp[r, c]=(m[r, c]-sum(k=c+1, r-1, tmp[r, k]*tmp[k, c])) /(tmp[c, c]+tmp[r, r]) ); ); return(tmp); } ff = exp(x)-1 Mff = matrix(6, 6, r, c, polcoeff(ff^(c-1), (r-1))) \\ create Bell-matrix for ff Mf = trisqrt ( Mff ) \\ = Mff^(1/2) is Bellmatrix for f f = Ser(Mf[, 2]) \\ coefficients of power series for half-iterate of exp(x)-1 from second column in Mf CROSSREFS Cf. A052122, A052123. Sequence in context: A009187 A009803 A009615 * A079194 A179198 A372390 Adjacent sequences: A184008 A184009 A184010 * A184012 A184013 A184014 KEYWORD sign AUTHOR Gottfried Helms, Feb 13 2011 STATUS approved

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Last modified September 18 08:43 EDT 2024. Contains 375998 sequences. (Running on oeis4.)