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 1-5 of 5 results.

A118206 Euler transform of the Liouville function.

Original entry on oeis.org

1, 1, 0, -1, 0, 0, 0, -1, -1, 0, 2, 0, -2, -2, 1, 2, 2, -2, -2, 0, 2, -1, -1, -2, 2, 5, 4, -5, -5, -2, 4, 2, -2, -7, 3, 8, 5, -7, -6, 1, 14, 4, -9, -14, 2, 5, 5, -10, -7, 6, 22, 3, -12, -20, 1, 15, 15, -16, -12, 4, 25, 6, -14, -31, 13, 33, 14, -39, -32, -6, 39, 15, -20, -31, 33, 41, 14, -53, -44, 3, 66, 12, -35, -51, 22, 48, 36, -60, -43, 21
Offset: 0

Views

Author

Stuart Clary, Apr 15 2006

Keywords

Links

Crossrefs

Cf. A118205, A118207, A118208, A061020, A117209.

Programs

  • Mathematica
    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 ]
max = 100; s = Product[(1 - x^k)^(-LiouvilleLambda[k]), {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d*moebius(core(d)))*x^m/m)+x*O(x^n)),n)} /* Cf. A061020 - Paul D. Hanna, Sep 22 2011 */

Formula

G.f.: A(x) = Product_{k>=1} (1 - x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.
Logarithmic derivative yields A061020. - Paul D. Hanna, Sep 22 2011
G.f.: A(x) = Product_{k >= 1} C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

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

Views

Author

Stuart Clary, Apr 15 2006

Keywords

Links

Crossrefs

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

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 ]
(* 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 *)

Formula

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)

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

A308396 Expansion of e.g.f. exp(-Sum_{k>=1} x^(k^2)/k^2).

Original entry on oeis.org

1, -1, 1, -1, -5, 29, -89, 209, 841, -50905, 458641, -2423521, 8243731, 158742869, -2450634185, 18519809489, -1402926535919, 21355930009679, -139305034406879, 306503668195775, 40578438892908331, -816475138658703091, 6941097158619626311, -24787202385366731311
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2019

Keywords

Crossrefs

Cf. A008836, A118205, A190585, A205801, A306831, A308397, A308398.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[-Sum[x^(k^2)/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

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

A356907 Expansion of 1 / (1 + Sum_{k>=1} lambda(k)*x^k), where lambda() is the Liouville function (A008836).

Original entry on oeis.org

1, -1, 2, -2, 2, 0, -4, 12, -22, 34, -42, 38, -6, -68, 202, -394, 616, -782, 730, -204, -1104, 3486, -6994, 11142, -14452, 14026, -5296, -17558, 60042, -123860, 201128, -266384, 268176, -124034, -273626, 1030396, -2188864, 3624290, -4898740, 5101306, -2744408
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A008836, A118205, A118206, A118207, A118208, A185694, A307076, A307240.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 + Sum[LiouvilleLambda[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[LiouvilleLambda[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 40}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A008836(k) * a(n-k).
Showing 1-5 of 5 results.

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