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

1, 2, 5, 10, 22, 42, 85, 162, 314, 588, 1113, 2066, 3847, 7080, 13036, 23824, 43504, 79048, 143441, 259376, 468313, 843352, 1516515, 2721470, 4877165, 8726118, 15593224, 27826634, 49602226, 88316198, 157089101, 279137436, 495566701, 879034448, 1557979289

Offset: 0

Author

Paul D. Hanna, Aug 10 2002

Keywords

• easy
• nice
• nonn

Comments

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

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

Related limits are limit_{n->infinity} a(n) r^n/n = 0.406...(?) and limit_{n->infinity} a(n+1)/a(n) = 1.723262561763844...

This sequence is the self-convolution of A073776.

Examples

			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,...}.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A073776, A068341, A070965, A008683.

Programs

Haskell

a073777 n = a073777_list !! (n-1)
a073777_list = 1 : f [1] where
   f xs = y : f (y : xs) where y = sum $ zipWith (*) xs ms'
   ms' = map negate $ tail a068341_list
-- Reinhard Zumkeller, Nov 03 2015

Mathematica

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 *)

Formula

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

Extensions

Corrected by Jean-François Alcover, Oct 10 2011