%I #35 May 10 2022 07:50:24
%S 1,1,2,3,6,9,17,28,50,83,147,249,435,742,1288,2207,3819,6561,11333,
%T 19497,33640,57915,99874,172020,296550,510886,880580,1517226,2614889,
%U 4505745,7765094,13380640,23059193,39735969,68476885,118001888
%N a(n) = Sum_{k=1..n} -mu(k+1) * a(n-k), with a(0)=1.
%C Recurrence relation involves the Moebius function.
%C Radius of convergence of A(x) is r=0.5802946238073267...
%C Related limits are
%C lim_{n->infinity} a(n) r^n = 0.6303632342... and
%C lim_{n->infinity} a(n+1)/a(n) = 1.723262561763844...
%C From _Gary W. Adamson_, Aug 11 2016: (Start)
%C The definition in the heading follows from the INVERTi transform of (1, 2, 3, 6, 9, 17, ...) equals -mu(n) for n >= 2 (cf. A157658).
%C Then for example, a(6) = 17 = (1, 1, 0, 1, -1, 1) dot (9, 6, 3, 2, 1, 1) = (9 + 6 + 0 + 2 - 1 + 1); in agreement with the first example. (End)
%H Reinhard Zumkeller, <a href="/A073776/b073776.txt">Table of n, a(n) for n = 0..1000</a>
%H Yu Hin (Gary) Au, <a href="https://arxiv.org/abs/2205.03680">Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series</a>, arXiv:2205.03680 [math.CO], 2022.
%F G.f.: A(x) = x / (Sum_{n>=1} mu(n)*x^n), A(0)=1, where mu(n) = Moebius function of n.
%F G.f. A(x) satisfies: x*A(x) = Sum_{n>=1} x^n*A(x)^n / A( x^n*A(x)^n ). - _Paul D. Hanna_, Apr 19 2016
%e a(6) = -mu(2)a(5) - mu(3)a(4) - mu(4)a(3) - mu(5)a(2) - mu(6)a(1) - mu(7)a(0) = 9 + 6 + 0 + 2 - 1 + 1 = 17.
%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 17*x^6 + 28*x^7 + 50*x^8 + 83*x^9 + 147*x^10 + 249*x^11 + 435*x^12 + ...
%e where
%e 1/A(x) = 1 - x - x^2 - x^4 + x^5 - x^6 + x^9 - x^10 - x^12 + x^13 + x^14 - x^16 - x^18 + x^20 + x^21 - x^22 + x^25 - x^28 - x^29 - x^30 + ... + mu(n)*x^n +...
%e Also, g.f. A(x) satisfies:
%e x*A(x) = x*A(x)/A(x*A(x)) + x^2*A(x)^2/A(x^2*A(x)^2) + x^3*A(x)^3/A(x^3*A(x)^3) + x^4*A(x)^4/A(x^4*A(x)^4) + x^5*A(x)^5/A(x^5*A(x)^5) + ...
%t a[0] = 1; a[n_] := a[n] = Sum[-MoebiusMu[k + 1]*a[n - k], {k, 1, n}]; Array[a,35,0] (* _Jean-François Alcover_, Apr 11 2011 *)
%o (Haskell)
%o a073776 n = a073776_list !! (n-1)
%o a073776_list = 1 : f [1] where
%o f xs = y : f (y : xs) where y = sum $ zipWith (*) xs ms
%o ms = map negate $ tail a008683_list
%o -- _Reinhard Zumkeller_, Nov 03 2015
%o (PARI) {a(n) = my(A=[1,1],F); for(i=1,n, A=concat(A,0); F=Ser(A); A = Vec(sum(m=1,#A, subst(x/F, x, x^m*F^m))) ); A[n+1]}
%o for(n=0,50, print1(a(n),", ")) \\ _Paul D. Hanna_, Apr 19 2016
%Y Cf. A073777, A068341, A070965, A008683, A185694, A157658.
%K easy,nice,nonn
%O 0,3
%A _Paul D. Hanna_, Aug 10 2002