A073776 a(n) = Sum_{k=1..n} -mu(k+1) * a(n-k), with a(0)=1.
1, 1, 2, 3, 6, 9, 17, 28, 50, 83, 147, 249, 435, 742, 1288, 2207, 3819, 6561, 11333, 19497, 33640, 57915, 99874, 172020, 296550, 510886, 880580, 1517226, 2614889, 4505745, 7765094, 13380640, 23059193, 39735969, 68476885, 118001888
Offset: 0
Examples
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. 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 + ... where 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 +... Also, g.f. A(x) satisfies: 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) + ...
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.
Programs
-
Haskell
a073776 n = a073776_list !! (n-1) a073776_list = 1 : f [1] where f xs = y : f (y : xs) where y = sum $ zipWith (*) xs ms ms = map negate $ tail a008683_list -- Reinhard Zumkeller, Nov 03 2015
-
Mathematica
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 *) -
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]} for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Apr 19 2016
Formula
G.f.: A(x) = x / (Sum_{n>=1} mu(n)*x^n), A(0)=1, where mu(n) = Moebius function of n.
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
Comments