A117208 -id:A117208 - cp's OEIS frontend

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

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

A117210 G.f. A(x) satisfies (1+x) = product_{n>=1} A(x^n).

Original entry on oeis.org

1, 1, -1, -2, 0, 1, 1, 0, -1, -1, 2, 1, -2, -3, 2, 4, 2, -5, -4, 0, 5, 2, 1, -5, -1, 2, 5, -5, -2, -2, 5, -1, 3, -6, 2, 0, 11, -6, -4, -10, 12, -1, 6, -13, 5, -8, 16, -8, 9, -13, 17, -17, 7, -21, 25, -10, 22, -29, 20, -24, 34, -24, 27, -44, 35, -32, 39, -52, 45, -39, 66, -53, 47, -76, 70, -55, 79, -98, 66, -84, 115, -89

Offset: 0

Paul D. Hanna, Mar 03 2006

Self-convolution inverse is A117211.

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

Cf. A117212 (l.g.f.), A117211 (inverse); variants: A117208, A117209.

nmax = 81; 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, -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
Weigh transform of A008683(n). - Vladeta Jovovic, Apr 20 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

A118205 Euler transform of the negative of the Liouville function.

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -2, 2, 0, -2, 3, -2, 1, 2, -3, 3, -2, 0, 3, -2, 3, -2, 0, 2, -2, 3, -1, 0, 1, -2, 5, 0, 0, 1, -2, 1, 1, 2, 0, 1, -2, 1, 4, -1, 4, -2, -3, 6, -2, 5, 6, -8, 6, -4, 2, 9, -8, 7, -4, -1, 11, -1, 5, 1, -8, 5, 2, 4, 7, -8, 4, 2, 1, 14, -2, 0, -1, -6, 19, 2, 5, 6, -15, 12, 1, 3, 18, -17, 1, 9, 0, 29, -4, -3, 4, -13, 14, 17, 2, 0, -4

Offset: 0

Stuart Clary, Apr 15 2006

Peter Bala, Cyclotomic polynomials and the generating functions of A118205 and A118206
Wikipedia, Cyclotomic polynomial
Wikipedia, Liouville function

Cf. A061020, A117208, A118206, A118207, A118208.

nmax = 100; 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 ]
(* Second program (needs Mma >= 7.0): *)
nmax = 100;
Product[(1 - x^n)^LiouvilleLambda[n], {n, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& (* Jean-François Alcover, Jan 08 2020 *)

G.f.: A(x) = Product_{k>=1} (1 - x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

G.f.: A(x) = - Product_{k >= 1} C(k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

A357521 Expansion of Product_{k>=1} (1 - mu(k)*x^k).

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -3, 3, -2, 0, 1, -2, 2, -2, 1, -2, 3, -4, 4, -2, 0, 2, -5, 6, -5, 3, -2, 1, -1, 1, 0, 0, 3, -6, 6, -5, 4, 0, -5, 7, -7, 5, -2, 2, 0, -2, 0, 1, 5, -7, 11, -14, 11, -6, -1, 9, -12, 8, -11, 11, -6, 10, -13, 8, -2, -12, 26, -26, 24, -20, 2, 11, -8, 14, -15, 9

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

sign

Cf. A008683, A117208, A185694, A292561, A300663, A306327, A357524, A357525.

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}]

A351403 G.f. A(x) satisfies: (1 - x) = Product_{i>=1, j>=1} A(x^(i*j)).

Original entry on oeis.org

1, -1, 2, 0, 0, 4, -5, 9, -6, 3, 4, -9, 15, -17, 13, -8, 0, 1, -9, 12, -17, 15, -25, 29, -27, 12, -3, -14, 28, -55, 63, -54, 53, -46, 18, 32, -57, 85, -106, 122, -108, 43, 8, -29, 80, -161, 148, -115, 104, -78, 57, 29, -77, 89, -99, 263, -283, 182, -212, 133, 49

Offset: 0

Ilya Gutkovskiy, Feb 10 2022

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Convolution inverse of A351402.

Cf. A000005, A007427, A101035, A117208, A288098, A351402.

nmax = 60; A007427[n_] := Sum[MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}]; CoefficientList[Series[Product[(1 - x^k)^A007427[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f. A(x) satisfies: (1 - x) = Product_{k>=1} A(x^k)^A000005(k).
G.f.: Product_{k>=1} (1 - x^k)^A007427(k).
G.f.: exp( -Sum_{k>=1} A101035(k) * x^k / k ).
a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A101035(k) * a(n-k).

A307656 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, 1, 0, 8, -7, 22, -6, 13, 29, -11, 82, -36, 114, 13, 103, 88, 88, 275, -20, 549, -200, 1007, -144, 811, 730, 188, 2093, -777, 3538, -643, 4083, -537, 4562, 2478, 1973, 8062, -3508, 17362, -8164, 20281, -2227, 17483, 8605, 2946, 30190, -6085, 53176, -28913, 78516

Offset: 0

Ilya Gutkovskiy, Apr 20 2019

sign

Convolution inverse of A307648.

			G.f.: A(x) = 1 - x + 2*x^2 + x^3 + 8*x^5 - 7*x^6 + 22 x^7 - 6*x^8 + 13*x^9 + 29*x^10 - 11*x^11 + 82*x^12 - 36*x^13 + ...

Cf. A008683, A046970, A055615, A117208, A307648.

terms = 50; CoefficientList[Series[Product[(1 - x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 50; CoefficientList[Series[Exp[-Sum[Sum[MoebiusMu[d] d^2, {d, Divisors[k]}] x^k/k, {k, 1, terms}]], {x, 0, terms}], x]
terms = 50; 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).

G.f.: exp(-Sum_{k>=1} A046970(k)*x^k/k).

cp's OEIS Frontend

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

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A117210 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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A118205 Euler transform of the negative of the Liouville function.

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A357521 Expansion of Product_{k>=1} (1 - mu(k)*x^k).

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A351403 G.f. A(x) satisfies: (1 - x) = Product_{i>=1, j>=1} A(x^(i*j)).

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A307656 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* ...

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A307656 G.f. A(x) satisfies: (1 - x) = A(x)A(x^2)^2A(x^3)^3A(x^4)^4 ... A(x^k)^k ...