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A307064 Expansion of 1 - 1/Sum_{k>=0} k!!*x^k.

%I A307064 #12 Jan 24 2024 10:10:00
%S A307064 0,1,1,0,3,1,18,13,155,168,1691,2381,22022,37401,331087,649036,
%T A307064 5626103,12372161,106486594,257573405,2220690451,5824952232,
%U A307064 50593271507,142387607469,1250521775454,3745193283657,33338037080183,105558942751948,953776675614223
%N A307064 Expansion of 1 - 1/Sum_{k>=0} k!!*x^k.
%H A307064 G. C. Greubel, <a href="/A307064/b307064.txt">Table of n, a(n) for n = 0..800</a>
%F A307064 a(0) = 0; a(n) = n!! - Sum_{k=1..n-1} k!!*a(n-k).
%t A307064 nmax = 28; CoefficientList[Series[1 - 1/Sum[k!! x^k, {k, 0, nmax}], {x, 0, nmax}], x]
%t A307064 a[0] = 0; a[n_]:= a[n] = n!! - Sum[k!! a[n-k], {k,n-1}];
%t A307064 Table[a[n], {n, 0, 28}]
%o A307064 (Magma)
%o A307064 m:=80;
%o A307064 F2:= func< n | n mod 2 eq 0 select Round(2^(n/2)*Gamma(n/2+1)) else Round( Gamma((n+3)/2)*Binomial(n+1, Floor((n+1)/2))/2^((n+1)/2) ) >;
%o A307064 R<x>:=PowerSeriesRing(Rationals(), m);
%o A307064 [0] cat Coefficients(R!( 1 - 1/(&+[F2(j)*x^j : j in [0..m+2]]) )); // _G. C. Greubel_, Jan 24 2024
%o A307064 (SageMath)
%o A307064 from sympy import factorial2
%o A307064 m=80;
%o A307064 def f(x): return 1 - 1/sum(factorial2(k)*x^k for k in range(m+1))
%o A307064 def A307063_list(prec):
%o A307064     P.<x> = PowerSeriesRing(QQ, prec)
%o A307064     return P( f(x) ).list()
%o A307064 A307063_list(m) # _G. C. Greubel_, Jan 24 2024
%Y A307064 Cf. A000698, A003319, A006882, A292778.
%K A307064 nonn
%O A307064 0,5
%A A307064 _Ilya Gutkovskiy_, Mar 21 2019