A117210 -id:A117210 - cp's OEIS frontend

A117212 Sum_{d|n} a(d)/d = (-1)^(n-1)/n for n>=1; equals the logarithmic g.f. of A117210.

1, -3, -2, 1, -4, 6, -6, 1, -2, 12, -10, -2, -12, 18, 8, 1, -16, 6, -18, -4, 12, 30, -22, -2, -4, 36, -2, -6, -28, -24, -30, 1, 20, 48, 24, -2, -36, 54, 24, -4, -40, -36, -42, -10, 8, 66, -46, -2, -6, 12, 32, -12, -52, 6, 40, -6, 36, 84, -58, 8, -60, 90, 12, 1, 48, -60, -66, -16, 44, -72, -70, -2, -72, 108, 8, -18

Offset: 1

Paul D. Hanna, Mar 03 2006

			For n=6, Sum_{d|6} a(d)/d = a(1)/1 + a(2)/2 + a(3)/3 + a(6)/6 = 1/1 - 3/2 - 2/3 + 6/6 = -1/6.

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A008683, A023900, A055615, A062157, A117210, A117211.

nmax = 72; Drop[ CoefficientList[ Series[ Sum[ MoebiusMu[k] k x^k/(1 + x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* Stuart Clary, Apr 15 2006 *)
f[p_, e_] := 1 - p; f[2, e_] := If[e == 1, -3, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2023 *)

a(n)=sumdiv(n,d,d*moebius(d))*[1,3,1,-1][(n-1)%4+1]

G.f.: Sum_{n>=1} a(n)*x^n/n = log(F(x)), where F(x) is the g.f. of A117210 and satisfies: (1+x) = Product_{n>=1} F(x^n).
a(n) = A023900(n) if n (mod 4) = 1 or 3, a(n) = 3*A023900(n) if n (mod 4) = 2, a(n) = -A023900(n) if n (mod 4) = 0, where A023900 is the Dirichlet inverse of Euler totient function.
From Stuart Clary, Apr 15 2006: (Start)
G.f.: A(x) = sum_{k>=1} mu(k) k x^k/(1 + x^k) where mu(k) is the Möbius function, A008683.
G.f.: A(x) is x times the logarithmic derivative of A117210(x).
G.f.: A(x) = A023900(x) - 2 A023900(x^2).
a(n) = sum_{d|n} (-1)^(n/d - 1) mu(d) d.
(End)
Dirichlet convolution of A055615 and A062157, so the Dirichlet g.f. is the product zeta(s)*(1-2^(1-s))/zeta(s-1). - R. J. Mathar, Feb 07 2011
Multiplicative with a(2) = -3, a(2^e) = 1 for e >= 2, and a(p^e) = 1 - p for an odd prime p. - Amiram Eldar, Aug 31 2023

A117209 G.f. A(x) satisfies 1/(1-x) = Product_{k>=1} A(x^k).

Original entry on oeis.org

1, 1, 0, -1, -1, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, 2, -1, -1, -2, 0, 1, 3, -1, 0, -1, 1, -1, 1, -3, 1, -1, 1, -2, 3, 0, 6, -1, -1, -6, 2, -4, 4, -3, 2, -4, 6, -5, 6, -2, 7, -5, 4, -13, 5, -3, 11, -6, 8, -14, 10, -6, 9, -14, 11, -14, 15, -13, 9, -15, 24, -13, 19, -21, 12, -20, 27, -24, 21, -26, 22, -24, 33, -33, 32, -26

Offset: 0

Paul D. Hanna, Mar 03 2006

sign

Self-convolution inverse is A117208.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)
N. J. A. Sloane, Transforms

Cf. A023900 (l.g.f.), A117208 (inverse); variants: A117210, A117211, A117212.

Cf. A008683.

nmax = 85; CoefficientList[ Series[ Product[ (1 - x^k)^(-MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)

{a(n)=polcoeff(exp(sum(k=1,n+1,sumdiv(k,d,d*moebius(d))*x^k/k)+x*O(x^n)),n)}

G.f.: A(x) = exp( Sum_{n>=1} A023900(n)*x^n/n ), where A023900 is the Dirichlet inverse of Euler totient function.
Euler transform of the Möbius function A008683. - Stuart Clary, Franklin T. Adams-Watters and Vladeta Jovovic, Apr 15 2006
G.f.: A(x) = Product_{k>=1}(1 - x^k)^(-mu(k)) where mu(k) is the Möbius function, A008683. - Stuart Clary and Franklin T. Adams-Watters, Apr 15 2006
G.f.: A(x) = Product_{k>=1} (1 + x^(2*k-1))^mu(2*k-1), where mu() is the Moebius function. - Seiichi Manyama, Jul 06 2024

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

A117208 G.f. A(x) satisfies (1-x) = product_{n>=1} A(x^n).

Original entry on oeis.org

1, -1, 1, 0, 0, 1, -1, 2, -1, 1, 0, 1, 0, 1, 0, 0, 2, -1, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 1, 0, 2, 0, 3, 0, 0, 2, 0, 3, 0, 3, -1, 2, 0, 4, 1, 1, 3, -3, 5, 1, 3, 0, 2, -1, 2, 4, 2, 4, -5, 6, -1, 2, 7, -2, 1, -1, 4, 3, 5, 2, -2, 1, 1, 8, 2, 4, -1, -3, 4, 9, 4, -2, 4, -7, 6, 7, 10, -1, -3, -1, 1, 11, 4, 8, -15, 2, 5, 7, 13, 1, -9, -7, 9

Offset: 0

Paul D. Hanna, Mar 03 2006

sign

Self-convolution inverse is A117209.

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A023900 (l.g.f.), A117209 (inverse); variants: A117210, A117211, A117212.

nmax = 106; CoefficientList[ Series[ Product[ (1 - x^k)^(MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)

{a(n)=polcoeff(exp(-sum(k=1,n+1,sumdiv(k,d,d*moebius(d))*x^k/k)+x*O(x^n)),n)}

G.f.: A(x) = exp( -Sum_{n>=1} A023900(n)*x^n/n ), where A023900 is the Dirichlet inverse of Euler totient function.
Euler transform of the negative of the Möbius function. - Stuart Clary, Apr 15 2006
G.f.: A(x) = product_{k>=1}(1 - x^k)^mu(k) where mu(k) is the Möbius function, A008683. - Stuart Clary, Apr 15 2006

A117211 G.f. A(x) satisfies 1/(1+x) = product_{n>=1} A(x^n).

Original entry on oeis.org

1, -1, 2, -1, 1, 1, -2, 4, -4, 4, -3, 2, 0, -1, 2, -3, 4, -5, 5, -4, 4, -3, 1, 1, -2, 3, -5, 5, -5, 3, -1, 1, 3, -4, 3, -2, 2, -1, -3, 4, -6, 4, -4, 5, 0, -4, 2, -1, 4, -2, 3, -3, 6, -9, 7, -1, 1, -4, -8, 10, -6, 10, -11, 12, -9, -4, 7, -7, 15, -25, 10, -5, 13, 1, -6, 16, -21, 14, -15, 28, -6, -12, -3, 1, 18, -18, 17, -25, 13

Offset: 0

Paul D. Hanna, Mar 03 2006

sign

Self-convolution inverse is A117210.

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A117212 (l.g.f.), A117210 (inverse); variants: A117208, A117209.

nmax = 88; CoefficientList[ Series[ Product[ (1 + x^k)^(-MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)

{a(n)=if(n==0,1,if(n==1,-1, (-1)^n-polcoeff(prod(i=1,n,sum(k=0,min(n\i,n-1),a(k)*x^(i*k))+x*O(x^n)),n,x)))}

G.f.: A(x) = exp( -Sum_{n>=1} A117212(n)*x^n/n ).

G.f.: A(x) = product_{k>=1}(1 + x^k)^(-mu(k)) where mu(k) is the Möbius function, A008683. - Stuart Clary, Apr 15 2006

A118207 Expansion of Product_{k>=1} (1 + x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

Original entry on oeis.org

1, 1, -1, -2, 1, 2, 0, -2, -2, 0, 5, 2, -7, -6, 7, 9, 0, -10, -9, 4, 17, 2, -18, -12, 14, 21, 5, -26, -25, 14, 41, 4, -38, -35, 18, 53, 23, -56, -54, 31, 86, 15, -78, -85, 34, 112, 41, -110, -102, 49, 158, 40, -138, -150, 68, 195, 68, -191, -190, 69, 279, 89, -217, -253, 102, 327, 122, -336, -335, 118, 462, 142, -361, -430, 170

Offset: 0

Stuart Clary, Apr 15 2006

Wikipedia, Cyclotomic polynomial
Wikipedia, Liouville function

Cf. A117210, A118205, A118206, A118208, A118209.

nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 + x^k)^lambda[k], {k, 1, nmax} ], {x, 0, nmax} ], x ]
(* From version 7 on *) nmax = 80; CoefficientList[ Series[ Product[ (1 + x^k)^LiouvilleLambda[k], {k, 1, nmax}], {x, 0, nmax}], x] (* Jean-François Alcover, Jul 30 2013 *)

From Peter Bala, Apr 05 2023: (Start)
G.f.: A(x) = Product_{k >= 1} C(k,x^(2*k)) / C(k,x^k) = Product_{k >= 1} C(2*k,x^k) / C(4*k,x^k) = -Product_{k >= 1} C(k,x^(2*k)) * C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial.
Conjecture: A(x^2) = Product_{k >= 1} C(k,x^k) * C(k,(-x)^k). (End)

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

A307649 G.f. A(x) satisfies: (1 + x) = A(x)A(x^2)^2A(x^3)^3A(x^4)^4 ... A(x^k)^k ...

Original entry on oeis.org

1, 1, -2, -5, 0, 4, 9, 2, -10, -21, 29, 15, -18, -80, 50, 59, 207, -228, -244, -315, 868, 103, 360, -1907, 752, -151, 3802, -5032, 965, -5279, 13742, -6049, 9107, -33835, 25398, -15098, 63365, -79614, 51752, -117194, 196980, -156321, 209085, -435223, 463497, -441950, 871202, -1146187, 1023944, -1704179

Offset: 0

Ilya Gutkovskiy, Apr 19 2019

sign

Weigh transform of A055615.

			G.f.: A(x) = 1 + x - 2*x^2 - 5*x^3 + 4*x^5 + 9*x^6 + 2*x^7 - 10*x^8 - 21*x^9 + 29*x^10 + 15*x^11 - 18*x^12 - 80*x^13 + ...

Cf. A008683, A055615, A117210, A307648.

terms = 49; CoefficientList[Series[Product[(1 + x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 49; A[] = 1; Do[A[x] = (1 + x)/Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} (1 + x^k)^(mu(k)*k).

A357525 Expansion of Product_{k>=1} (1 + mu(k)*x^k).

Original entry on oeis.org

1, 1, -1, -2, -1, 0, 1, 1, 0, 0, 1, 0, -2, -2, 1, 4, 3, -2, -4, -2, 0, 2, 3, 0, -1, 1, 0, -3, -3, -1, 2, 4, 3, 0, -2, -1, 2, 0, -5, -3, 3, 3, 0, -2, -4, -2, 4, 5, 3, 3, 1, -4, -9, -8, 3, 11, 6, 0, -3, -7, -4, 2, -1, -2, 6, 8, -2, -10, -8, 4, 14, 11, 2, -6, -11, -5

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

sign

Cf. A008683, A087188, A117210, A185694, A300663, A306327, A357521, A357524.

nmax = 75; CoefficientList[Series[Product[(1 + MoebiusMu[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Sum[d (-MoebiusMu[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A307658 G.f. A(x) satisfies: (1 + x)/(1 - x) = A(x)A(x^2)A(x^3)A(x^4) ... A(x^k) ...

Original entry on oeis.org

1, 2, 0, -4, -4, 0, 4, 4, 0, -4, 0, 4, 0, -8, -4, 8, 16, 0, -20, -20, 8, 24, 20, -12, -24, -8, 24, 4, -16, -24, 16, 28, 24, -40, -32, 0, 72, 24, -28, -104, 0, 48, 88, -44, -32, -64, 92, 20, 24, -124, 64, 0, 96, -168, -12, -72, 272, -24, 72, -300, 104, -88, 316, -272, 128, -272, 376, -300

Offset: 0

Ilya Gutkovskiy, Apr 20 2019

sign

Convolution of A117209 and A117210.

			G.f.: A(x) = 1 + 2*x - 4*x^3 - 4*x^4 + 4*x^6 + 4*x^7 - 4*x^9 + 4*x^11 - 8*x^13 - 4*x^14 + 8*x^15 + ...

Cf. A008683, A117209, A117210, A307659.

terms = 67; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^MoebiusMu[k], {k, 1, terms}], {x, 0, terms}], x]
terms = 67; A[] = 1; Do[A[x] = (1 + x)/((1 - x) Product[A[x^k], {k, 2, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^mu(k).

cp's OEIS Frontend

A117212 Sum_{d|n} a(d)/d = (-1)^(n-1)/n for n>=1; equals the logarithmic g.f. of A117210.

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A117209 G.f. A(x) satisfies 1/(1-x) = Product_{k>=1} A(x^k).

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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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A117208 G.f. A(x) satisfies (1-x) = product_{n>=1} A(x^n).

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A117211 G.f. A(x) satisfies 1/(1+x) = product_{n>=1} A(x^n).

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A118207 Expansion of Product_{k>=1} (1 + x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

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

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A307649 G.f. A(x) satisfies: (1 + x) = A(x)*A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

A307649 G.f. A(x) satisfies: (1 + x) = A(x)A(x^2)^2A(x^3)^3A(x^4)^4 ... A(x^k)^k ...

A307658 G.f. A(x) satisfies: (1 + x)/(1 - x) = A(x)A(x^2)A(x^3)A(x^4) ... A(x^k) ...