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%I #63 Sep 08 2022 08:45:48
%S 1,-1,-1,-3,-13,-71,-461,-3447,-29093,-273343,-2829325,-31998903,
%T -392743957,-5201061455,-73943424413,-1123596277863,-18176728317413,
%U -311951144828863,-5661698774848621,-108355864447215063
%N Expansion of g.f.: 1/(Sum_{k >= 0} k!*x^k).
%C Essentially the same as A003319, which is the main entry for these numbers. - _N. J. A. Sloane_, Jun 11 2013
%D M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 40.
%H G. C. Greubel, <a href="/A167894/b167894.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) = -Sum_{i=1}^{n-1} a(i)*(n-i)!, n>1, a(0)=0, a(1)=1. - _Vladimir Kruchinin_, Aug 09 2010
%F 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:
%F G.f.: 1 - x/Q(0), where Q(k) = 1 - (k+1)*x/(1 - (k+2)*x/Q(k+1)).
%F G.f.: U(0) where U(k) = 1 - x*(k+1)/(1 - x*(k+1)/U(k+1)).
%F 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))).
%F G.f.: A(x) = 1 - x/G(0) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1).
%F G.f.: x*Q(0), where Q(k) = 1/x - 1 - 2*k - (k+1)^2/Q(k+1).
%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))).
%F G.f.: 2/Q(0), where Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1) )).
%F G.f.: conjecture: Q(0), where Q(k) = 1 + k*x - (k+1)*x/Q(k+1). (End)
%F a(n) ~ -n! * (1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - 25912/n^7 - 319339/n^8 - 4388949/n^9 - 66495386/n^10 - ...). - _Vaclav Kotesovec_, Dec 08 2020
%t CoefficientList[Series[1/(Sum[k!*x^k, {k, 0, 25}]), {x, 0, 20}], x] (* _G. C. Greubel_, Jun 30 2016 *)
%o (Maxima) a(n):=if n=1 then 1 else -sum((n-i)!*a(i),i,1,n-1); \\ _Vladimir Kruchinin_, Aug 09 2010
%o (Sage)
%o def A167894_list(len):
%o R, C = [1], [1]+[0]*(len-1)
%o for n in (1..len-1):
%o for k in range(n, 0, -1):
%o C[k] = C[k-1] * k
%o C[0] = -sum(C[k] for k in (1..n))
%o R.append(C[0])
%o return R
%o print(A167894_list(20)) # _Peter Luschny_, Feb 19 2016
%o (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
%o (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
%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(&+[Factorial(k)*x^k: k in [0..m+1]]) )); // _G. C. Greubel_, Feb 07 2019
%Y Cf. A003319, A158882.
%K sign
%O 0,4
%A _Philippe Deléham_, Nov 15 2009
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