A068341 -id:A068341 - cp's OEIS frontend

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

Paul D. Hanna, Aug 10 2002

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.

			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,...}.

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

Cf. A073776, A068341, A070965, A008683.

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

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 *)

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

Corrected by Jean-François Alcover, Oct 10 2011

A073776 a(n) = Sum_{k=1..n} -mu(k+1) * a(n-k), with a(0)=1.

Original entry on oeis.org

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

Paul D. Hanna, Aug 10 2002

Recurrence relation involves the Moebius function.
Radius of convergence of A(x) is r=0.5802946238073267...
Related limits are
  lim_{n->infinity} a(n) r^n = 0.6303632342... and
  lim_{n->infinity} a(n+1)/a(n) = 1.723262561763844...
From Gary W. Adamson, Aug 11 2016: (Start)
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).
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)

			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) + ...

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.

Cf. A073777, A068341, A070965, A008683, A185694, A157658.

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

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 *)

{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

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

A300663 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 0, -2, -3, -2, 3, 8, 8, -2, -16, -24, -10, 24, 59, 54, -11, -117, -174, -90, 162, 431, 449, -20, -835, -1393, -848, 1062, 3352, 3748, 317, -6257, -11134, -7583, 7294, 25956, 30786, 5217, -46545, -88132, -65062, 48534, 199234, 249263, 63034, -342174, -691679, -554002

Offset: 0

Ilya Gutkovskiy, Mar 10 2018

sign

Invert transform of A008683.

Seiichi Manyama, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms

Cf. A008683, A050385, A068341, A073776, A073777, A117209, A117210, A185694, A280194.

a:= proc(n) option remember; `if`(n=0, 1, add(
      numtheory[mobius](j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 10 2018

nmax = 47; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[MoebiusMu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 47}]

my(N=66, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, moebius(k)*x^k))) \\ Seiichi Manyama, Apr 06 2022

a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*a(n-k))); \\ Seiichi Manyama, Apr 06 2022

G.f.: 1/(1 - Sum_{k>=1} A008683(k)*x^k).

a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * a(n-k). - Seiichi Manyama, Apr 06 2022

A112966 Sum(mu(i)*omega(j): i+j=n), with mu=A008683 and omega=A001221.

Original entry on oeis.org

0, 0, 1, 0, -1, -1, -1, -2, -3, -2, -2, -1, -3, -3, -4, 0, -6, -1, -4, -1, -3, -3, -7, 0, -5, -3, -3, 0, -5, 1, -5, -2, -10, -1, -8, 4, -8, -3, -4, 2, -6, 0, -5, -1, -4, -2, -11, 3, -8, -1, -8, -1, -11, 2, -8, 2, -7, -3, -9, 5, -2, -5, -7, 2, -11, 7, -6, 0, -4, 1, -9, 4, -12, -3, -6, 0, -10, 2, -7, -1, -10, -8, -12, 6, -13, -2, -12, 0

Offset: 1

Reinhard Zumkeller, Oct 07 2005

			a(5)=mu(1)*omega(4)+mu(2)*omega(3)+mu(3)*omega(2)+mu(4)*omega(1)
= 1*1 - 1*1 - 1*1 + 0*0 = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A013939, A112965, A068341, A112962, A112963, A112964, A112968.

a112966 n = sum $ zipWith (*)
   a008683_list $ reverse $ take (n - 1) a001221_list
-- Reinhard Zumkeller, Feb 29 2012

A112968 a(n) = Sum_{i+j=n} mu(i)*Omega(j), with mu=A008683 and Omega=A001222.

Original entry on oeis.org

0, 0, 1, 0, 0, -2, -2, -2, -2, -2, -6, -2, -4, -2, -7, -1, -5, 0, -7, -3, -9, 1, -11, 2, -7, 1, -12, 1, -11, 7, -8, -5, -8, -1, -18, 3, -10, 1, -13, 1, -7, 13, -12, -2, -13, 6, -16, 3, -11, 3, -15, -4, -16, 13, -15, -4, -15, 4, -17, 11, -14, 4, -13, 7, -12, 15, -17, -5, -15, 16, -13, 3, -12, 3, -20, 3, -27, 19, -20, -3, -11, 3

Offset: 1

Reinhard Zumkeller, Oct 07 2005

sign

			a(5) = mu(1)*Omega(4)+mu(2)*Omega(3)+mu(3)*Omega(2)+mu(4)*Omega(1) = 1*2 - 1*1 - 1*1 + 0*1 = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A013939, A112967, A068341, A112962, A112963, A112964, A112966.

a112968 n = sum $ zipWith (*)
   a008683_list $ reverse $ take (n - 1) a001222_list
-- Reinhard Zumkeller, Feb 29 2012

A112968[n_]:=Plus@@Table[MoebiusMu[i]*PrimeOmega[n-i],{i,1,n-1}]; Array[A112968,200] (* Enrique Pérez Herrero, Feb 28 2012 *)

Corrected by N. J. A. Sloane, Mar 01 2006

A112963 Sum(mu(i)*tau(j): i+j=n), with mu=A008683 and tau=A000005.

Original entry on oeis.org

0, 1, 1, -1, -1, -4, -2, -5, -5, -4, -7, -4, -6, -7, -11, 0, -12, -1, -11, -6, -12, -1, -20, 2, -13, -2, -16, 2, -19, 9, -18, -9, -20, 4, -31, 10, -21, -2, -18, 7, -20, 14, -26, -3, -16, 13, -40, 5, -26, 7, -22, -1, -40, 18, -32, 2, -21, 10, -40, 16, -25, 5, -21, 17, -41, 31, -40, -4, -14, 30, -38, 3, -39, 8, -21, 14, -58

Offset: 1

Reinhard Zumkeller, Oct 07 2005

sign

			a(5)=mu(1)*tau(4)+mu(2)*tau(3)+mu(3)*tau(2)+mu(4)*tau(1)
= 1*3 - 1*2 - 1*2 + 0*1 = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006218, A055507, A068341, A112962, A112964, A112966, A112968.

a112963 n = sum $ zipWith (*)
   a008683_list $ reverse $ take (n - 1) a000005_list
-- Reinhard Zumkeller, Feb 29 2012

A112964 Sum(mu(i)*sigma(j): i+j=n), with mu=A008683 and sigma=A000203.

Original entry on oeis.org

0, 1, 2, 0, 0, -6, -3, -12, -11, -13, -22, -19, -20, -30, -41, -15, -55, -24, -52, -41, -59, -24, -109, -22, -78, -42, -111, -26, -131, -2, -119, -75, -133, -8, -214, 7, -175, -68, -176, -17, -209, 14, -231, -73, -175, 45, -349, -11, -236, -20, -236, -53, -384, 68, -321, -56, -270, 1, -457, 41, -328, -48

Offset: 1

Reinhard Zumkeller, Oct 07 2005

sign

			a(5)=mu(1)*sigma(4)+mu(2)*sigma(3)+mu(3)*sigma(2)+mu(4)*sigma(1)
= 1*7 - 1*4 - 1*3 + 0*1 = 0.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A024916, A000385, A068341, A112962, A112963, A112966, A112968.

a(n)=sum(i=1,n-1,moebius(i)*sigma(n-i)) \\ Charles R Greathouse IV, Feb 14 2013

A112962 Sum(mu(i)*phi(j): i+j=n), with mu=A008683 and phi=A000010.

Original entry on oeis.org

0, 1, 0, 0, -1, -1, -4, -2, -5, -8, -5, -8, -9, -11, -10, -24, 1, -21, -11, -23, -15, -37, 4, -42, -11, -38, -7, -49, 6, -63, -12, -44, -3, -81, 10, -106, 7, -49, -8, -92, 15, -103, 2, -72, -5, -114, 41, -140, -3, -114, 8, -113, 49, -179, 3, -135, 27, -131, 46, -218, -7, -99, 32, -185, 72, -259, 50, -104, 23, -211, 52, -248, 43, -153

Offset: 1

Reinhard Zumkeller, Oct 07 2005

sign

			a(5)=mu(1)*phi(4)+mu(2)*phi(3)+mu(3)*phi(2)+mu(4)*phi(1) = 1*2 - 1*2 - 1*1 + 0*1 = -1.

Cf. A002088, A065093, A068341, A112963, A112964, A112966, A112968.

with(numtheory); f:=n->add(phi(i)*mobius(n-i),i=1..n-1);

a(n)=sum(i=1,n-1,moebius(i)*eulerphi(n-i)) \\ Charles R Greathouse IV, Feb 21 2013

Corrected by N. J. A. Sloane, Mar 01 2006

A300673 Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 0, -3, -6, 5, 61, 126, -308, -2772, -5669, 25630, 224730, 486551, -3068155, -29264219, -72173176, 513535711, 5625869262, 16687752839, -113740116822, -1496118902963, -5508392724427, 31534346503605, 523333047780288, 2414704077547660, -10254467367668159

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

sign

Exponential transform of A008683.

			E.g.f.: A(x) = 1 + x/1! - 3*x^3/3! - 6*x^4/4! + 5*x^5/5! + 61*x^6/6! + 126*x^7/7! - 308*x^8/8! - 2772*x^9/9! - 5669*x^10/10! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..592
N. J. A. Sloane, Transforms

Cf. A008683, A050385, A050386, A068341, A073776, A104688, A117209, A117210, A195589, A204290, A300663.

nmax = 26; CoefficientList[Series[Exp[Sum[MoebiusMu[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[MoebiusMu[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]

a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Feb 27 2022

E.g.f.: exp(Sum_{k>=1} A008683(k)*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Feb 27 2022

A341635 a(n) = Sum_{d|n} phi(d) * mu(d) * mu(n/d).

Original entry on oeis.org

1, -2, -3, 1, -5, 6, -7, 0, 2, 10, -11, -3, -13, 14, 15, 0, -17, -4, -19, -5, 21, 22, -23, 0, 4, 26, 0, -7, -29, -30, -31, 0, 33, 34, 35, 2, -37, 38, 39, 0, -41, -42, -43, -11, -10, 46, -47, 0, 6, -8, 51, -13, -53, 0, 55, 0, 57, 58, -59, 15, -61, 62, -14, 0, 65

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Dirichlet inverse of A003967.
Moebius transform of A097945.
From Vaclav Kotesovec, Feb 19 2021: (Start)
Abs(a(n)) <= n.
a(n) = n iff n is in A030229. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A003967, A007427, A007431, A008683, A030229 (fixed points), A046099 (positions of 0's), A068341, A097945, A276833.

Table[Sum[EulerPhi[d] MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 65}]
Table[Sum[MoebiusMu[GCD[n, k]] MoebiusMu[n/GCD[n, k]], {k, n}], {n, 65}]

a(n) = sumdiv(n, d, eulerphi(d)*moebius(d)*moebius(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} mu(gcd(n,k)) * mu(n/gcd(n,k)).
a(1) = 1; a(n) = -Sum_{d|n, d < n} A003967(n/d) * a(d).
a(n) = Sum_{d|n} mu(n/d) * A097945(d).
Multiplicative with a(p^e) = -p if e=1, p-1 if e=2, and 0 otherwise. - Amiram Eldar, Feb 19 2021

cp's OEIS Frontend

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

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A073776 a(n) = Sum_{k=1..n} -mu(k+1) * a(n-k), with a(0)=1.

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A300663 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683).

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A112966 Sum(mu(i)*omega(j): i+j=n), with mu=A008683 and omega=A001221.

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A112968 a(n) = Sum_{i+j=n} mu(i)*Omega(j), with mu=A008683 and Omega=A001222.

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A112963 Sum(mu(i)*tau(j): i+j=n), with mu=A008683 and tau=A000005.

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A112964 Sum(mu(i)*sigma(j): i+j=n), with mu=A008683 and sigma=A000203.

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PARI

A112962 Sum(mu(i)*phi(j): i+j=n), with mu=A008683 and phi=A000010.

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