%I #20 Dec 26 2023 22:53:30

%S 1,2,5,10,22,42,85,162,314,588,1113,2066,3847,7080,13036,23824,43504,

%T 79048,143441,259376,468313,843352,1516515,2721470,4877165,8726118,

%U 15593224,27826634,49602226,88316198,157089101,279137436,495566701,879034448,1557979289

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

%C Recurrence relation involves the convolution of the Moebius function (A068341).

%C Radius of convergence of A(x) is r=0.5802946238073267...

%C 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 This sequence is the self-convolution of A073776.

%H Reinhard Zumkeller, <a href="/A073777/b073777.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x)= x/(Sum_{n=1..infinity} mu(n)*x^n)^2, A(0)=1, where mu(n)=Moebius function.

%e 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 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 (Haskell)

%o a073777 n = a073777_list !! (n-1)

%o a073777_list = 1 : f [1] where

%o f xs = y : f (y : xs) where y = sum $ zipWith (*) xs ms'

%o ms' = map negate $ tail a068341_list

%o -- _Reinhard Zumkeller_, Nov 03 2015

%Y Cf. A073776, A068341, A070965, A008683.

%K easy,nice,nonn

%O 0,2

%A _Paul D. Hanna_, Aug 10 2002

%E Corrected by _Jean-François Alcover_, Oct 10 2011