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 A167894 Expansion of g.f.: 1/(Sum_{k >= 0} k!*x^k). 5
 1, -1, -1, -3, -13, -71, -461, -3447, -29093, -273343, -2829325, -31998903, -392743957, -5201061455, -73943424413, -1123596277863, -18176728317413, -311951144828863, -5661698774848621, -108355864447215063 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Essentially the same as A003319, which is the main entry for these numbers. - N. J. A. Sloane, Jun 11 2013 REFERENCES M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 40. LINKS G. C. Greubel, Table of n, a(n) for n = 0..400 FORMULA a(n) = -Sum_{i=1}^{n-1} a(i)*(n-i)!, n>1, a(0)=0, a(1)=1. - Vladimir Kruchinin, Aug 09 2010 From Sergei N. Gladkovskii, Jun 24 2012, Oct 15 2012, Nov 18 2012, Dec 26 2012, Apr 25 2013, May 29 2013, Aug 08 2013, Nov 19 2013: (Start) Continued fractions: G.f.: 1 - x/Q(0), where Q(k) = 1 - (k+1)*x/(1 - (k+2)*x/Q(k+1)). G.f.: U(0) where U(k) =  1 - x*(k+1)/(1 - x*(k+1)/U(k+1)). G.f.: 1/G(0) where G(k) = 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))). G.f.: A(x) = 1 - x/G(0) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1). G.f.: x*Q(0), where Q(k) = 1/x - 1 - 2*k - (k+1)^2/Q(k+1). G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))). G.f.: 2/Q(0), where Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1) )). G.f.: conjecture: Q(0), where Q(k) = 1 + k*x - (k+1)*x/Q(k+1). (End) MATHEMATICA CoefficientList[Series[1/(Sum[k!*x^k, {k, 0, 25}]), {x, 0, 20}], x] (* G. C. Greubel, Jun 30 2016 *) PROG (Maxima) a(n):=if n=1 then 1 else -sum((n-i)!*a(i), i, 1, n-1); \\ Vladimir Kruchinin, Aug 09 2010 (Sage) def A167894_list(len):     R, C = [1], [1]+[0]*(len-1)     for n in (1..len-1):         for k in range(n, 0, -1):             C[k] = C[k-1] * k         C[0] = -sum(C[k] for k in (1..n))         R.append(C[0])     return R print A167894_list(20) # Peter Luschny, Feb 19 2016 (Sage) m=20; (1/sum(factorial(k)*x^k for k in range(m+1))).series(x, m).coefficients(x, sparse=False) # G. C. Greubel, Feb 07 2019 (PARI) m=20; my(x='x+O('x^m)); Vec(1/sum(k=0, m+1, k!*x^k)) \\ G. C. Greubel, Feb 07 2019 (MAGMA) m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(&+[Factorial(k)*x^k: k in [0..m+1]]) )); // G. C. Greubel, Feb 07 2019 CROSSREFS Cf. A003319, A158882. Sequence in context: A122455 A126390 A272428 * A158882 A233824 A003319 Adjacent sequences:  A167891 A167892 A167893 * A167895 A167896 A167897 KEYWORD sign AUTHOR Philippe Deléham, Nov 15 2009 STATUS approved

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Last modified October 15 13:38 EDT 2019. Contains 328030 sequences. (Running on oeis4.)