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A073777 a(n) = Sum_{k=1..n} -A068341(k+1)*a(n-k), a(0)=1.

%I A073777 #20 Dec 26 2023 22:53:30
%S A073777 1,2,5,10,22,42,85,162,314,588,1113,2066,3847,7080,13036,23824,43504,
%T A073777 79048,143441,259376,468313,843352,1516515,2721470,4877165,8726118,
%U A073777 15593224,27826634,49602226,88316198,157089101,279137436,495566701,879034448,1557979289
%N A073777 a(n) = Sum_{k=1..n} -A068341(k+1)*a(n-k), a(0)=1.
%C A073777 Recurrence relation involves the convolution of the Moebius function (A068341).
%C A073777 Radius of convergence of A(x) is r=0.5802946238073267...
%C A073777 Related limits are limit_{n->infinity} a(n) r^n/n = 0.406...(?) and limit_{n->infinity} a(n+1)/a(n) = 1.723262561763844...
%C A073777 This sequence is the self-convolution of A073776.
%H A073777 Reinhard Zumkeller, <a href="/A073777/b073777.txt">Table of n, a(n) for n = 0..1000</a>
%F A073777 G.f.: A(x)= x/(Sum_{n=1..infinity} mu(n)*x^n)^2, A(0)=1, where mu(n)=Moebius function.
%e A073777 a(4) = -A068341(2)*a(3) -A068341(3)*a(2) -A068341(4)*a(1) -A068341(5)*a(0) = 2*10 +1*5 -2*2 +1*1 = 22. A068341 begins {1,-2,-1,2,-1,4,-2,0,3,...}.
%t A073777 A068341[n_] := A068341[n] = Sum[MoebiusMu[k]*MoebiusMu[n + 1 - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[-A068341[k + 1]*a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}] (* _Jean-François Alcover_, Oct 10 2011 *)
%o A073777 (Haskell)
%o A073777 a073777 n = a073777_list !! (n-1)
%o A073777 a073777_list = 1 : f [1] where
%o A073777    f xs = y : f (y : xs) where y = sum $ zipWith (*) xs ms'
%o A073777    ms' = map negate $ tail a068341_list
%o A073777 -- _Reinhard Zumkeller_, Nov 03 2015
%Y A073777 Cf. A073776, A068341, A070965, A008683.
%K A073777 easy,nice,nonn
%O A073777 0,2
%A A073777 _Paul D. Hanna_, Aug 10 2002
%E A073777 Corrected by _Jean-François Alcover_, Oct 10 2011