cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Paul D. Hanna, Mar 03 2006

Keywords

Comments

Self-convolution inverse is A117208.

Links

Crossrefs

Cf. A023900 (l.g.f.), A117208 (inverse); variants: A117210, A117211, A117212.
Cf. A008683.

Programs

  • Mathematica
    nmax = 85; CoefficientList[ Series[ Product[ (1 - x^k)^(-MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)
  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n+1,sumdiv(k,d,d*moebius(d))*x^k/k)+x*O(x^n)),n)}

Formula

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

Views

Author

Paul D. Hanna, Mar 03 2006

Keywords

Comments

Self-convolution inverse is A117211.

Links

Crossrefs

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

Programs

  • Mathematica
    nmax = 81; CoefficientList[ Series[ Product[ (1 + x^k)^(MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)
  • PARI
    {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)))}

Formula

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

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

Views

Author

Paul D. Hanna, Mar 03 2006

Keywords

Comments

Self-convolution inverse is A117209.

Links

Crossrefs

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

Programs

  • Mathematica
    nmax = 106; CoefficientList[ Series[ Product[ (1 - x^k)^(MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] (* Stuart Clary, Apr 15 2006 *)
  • PARI
    {a(n)=polcoeff(exp(-sum(k=1,n+1,sumdiv(k,d,d*moebius(d))*x^k/k)+x*O(x^n)),n)}

Formula

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

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Mar 03 2006

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

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

A118208 G.f.: A(x) = Product_{k>=1} (1 + x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.

Original entry on oeis.org

1, -1, 2, -1, 0, 2, -4, 5, -3, 0, 4, -6, 6, -2, -3, 8, -10, 6, 0, -6, 14, -13, 9, 0, -12, 17, -18, 11, 3, -18, 28, -22, 14, 7, -25, 30, -31, 11, 12, -23, 34, -28, 9, 12, -30, 35, -31, 10, 11, -30, 56, -35, 26, -4, -41, 51, -65, 48, -8, -28, 65, -74, 70, -9, -49, 71, -112, 69, -4, -48, 135, -129, 82, -21, -83, 155, -176, 99, 0
Offset: 0

Views

Author

Stuart Clary, Apr 15 2006

Keywords

Links

Crossrefs

Cf. A118205, A118206, A118207, A118209, A117211.

Programs

  • Mathematica
    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 ]

Formula

G.f.: A(x) = Product_{k >= 1} C(k,x^k)*C(2*k,x^(2*k)), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

A357524 Expansion of Product_{k>=1} 1 / (1 + mu(k)*x^k).

Original entry on oeis.org

1, -1, 2, -1, 2, 0, 1, 2, 0, 3, 0, 4, 1, 4, 2, 4, 4, 4, 5, 6, 6, 6, 8, 8, 10, 9, 11, 12, 13, 14, 17, 17, 20, 19, 23, 24, 28, 27, 30, 34, 34, 40, 41, 47, 48, 50, 56, 62, 64, 71, 72, 80, 85, 91, 99, 104, 113, 112, 128, 135, 147, 153, 159, 176, 180, 196, 210, 220, 233, 240, 264
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A008683, A117211, A185694, A300663, A306327, A329069, A357521, A357525.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[1/(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, 70}]

A307657 G.f. A(x) satisfies: 1/(1 + x) = A(x)*A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

Original entry on oeis.org

1, -1, 3, 0, 1, 10, -13, 33, -24, 26, 22, -53, 148, -181, 233, -146, 69, 157, -373, 721, -884, 1071, -1076, 1041, -444, -425, 1767, -3225, 5232, -7144, 7942, -6920, 4999, -1112, -5947, 15481, -24273, 32103, -39255, 43839, -37560, 18235, 6158, -35079, 81689, -134919, 167667, -179256
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 20 2019

Keywords

Comments

Convolution inverse of A307649.

Examples

			G.f.: A(x) = 1 - x + 3*x^2 + x^4 + 10*x^5 - 13*x^6 + 33*x^7 - 24*x^8 + 26*x^9 + 22*x^10 - 53*x^11 + 148*x^12 - 181*x^13 + ...

Crossrefs

Cf. A008683, A055615, A117211, A307649.

Programs

  • Mathematica
    terms = 47; CoefficientList[Series[Product[1/(1 + x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 47; A[] = 1; Do[A[x] = 1/((1 + x) Product[A[x^k]^k, {k, 2, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=1} 1/(1 + x^k)^(mu(k)*k).
Showing 1-7 of 7 results.

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