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 1-3 of 3 results.

A280128 Expansion of Product_{k>=2} (1 + mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 3, 3, 5, 4, 6, 7, 7, 9, 9, 11, 13, 14, 16, 19, 18, 24, 23, 28, 31, 33, 39, 42, 46, 52, 57, 63, 71, 76, 87, 92, 103, 113, 123, 135, 149, 161, 178, 193, 210, 231, 249, 274, 298, 323, 352, 382, 414, 451, 486, 528, 572, 617, 669
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 26 2016

Keywords

Comments

Number of partitions of n into distinct squarefree parts > 1 (A144338).

Examples

			G.f. = 1 + x^2 + x^3 + 2*x^5 + x^6 + 2*x^7 + 2*x^8 + 2*x^9 + 3*x^10 + 3*x^11 + ...
a(10) = 3 because we have [10], [7, 3] and [5, 3, 2].

Links

Crossrefs

Cf. A005117, A008683, A087188, A144338, A280127.

Programs

  • Maple
    with(numtheory): seq(coeff(series(mul(1+mobius(k)^2*x^k,k=2..n), x,n+1),x,n),n=0..70); # Muniru A Asiru, Jul 30 2018
  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1 + MoebiusMu[k]^2 x^k, {k, 2, nmax}], {x, 0, nmax}], x]
  • PARI
    {a(n) = if(n < 0, 0, polcoeff( prod(k=2, n, 1 + issquarefree(k)*x^k + x*O(x^n)), n))}; /* Michael Somos, Dec 26 2016 */

Formula

G.f.: Product_{k>=2} (1 + mu(k)^2*x^k).

A280169 Expansion of Product_{k>=2} 1/(1 - mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 8, 9, 10, 11, 13, 14, 17, 18, 21, 24, 26, 30, 33, 38, 42, 47, 53, 58, 65, 73, 80, 90, 99, 110, 122, 134, 149, 164, 181, 199, 220, 242, 266, 292, 321, 352, 386, 424, 463, 507, 554, 606, 662, 722, 788, 860, 936, 1020, 1111, 1208, 1314, 1428, 1553, 1685, 1829, 1984, 2152
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2016

Keywords

Comments

Number of partitions of n into odd squarefree parts > 1.

Examples

			a(13) = 3 because we have [13], [7, 3, 3] and [5, 5, 3].

Links

Crossrefs

Cf. A005117, A008683, A056911, A073576, A134345, A144338, A280127.

Programs

  • Mathematica
    nmax = 76; CoefficientList[Series[Product[1/(1 - MoebiusMu[2 k - 1]^2 x^(2 k - 1)), {k, 2, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=2} 1/(1 - mu(2*k-1)^2*x^(2*k-1)).

A280197 Expansion of 1/(1 - Sum_{k>=2} mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 6, 8, 12, 20, 28, 45, 68, 102, 159, 238, 367, 557, 849, 1298, 1973, 3015, 4592, 7002, 10679, 16276, 24822, 37841, 57696, 87971, 134119, 204497, 311783, 475370, 724786, 1105053, 1684853, 2568837, 3916642, 5971587, 9104711, 13881698, 21165024, 32269721, 49200718, 75014949, 114373158, 174381511
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2016

Keywords

Comments

Number of compositions (ordered partitions) into squarefree parts > 1 (A144338).

Examples

			a(5) = 3 because we have [5], [3, 2] and [2, 3].

Links

Crossrefs

Cf. A005117, A008683, A073576, A144338, A280127, A280194.

Programs

  • Maple
    N:= 100: # for a(0)..a(N)
g:= 1/(1-add(numtheory:-mobius(k)^2*x^k, k=2..N)):
S:= series(g,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Dec 29 2016
  • Mathematica
    nmax = 48; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=2} mu(k)^2*x^k).
Showing 1-3 of 3 results.

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