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 1-2 of 2 results.

A228156 Expansion of sqrt((1+4*x)/AGM(1+4*x,1-4*x)) where AGM denotes the arithmetic-geometric mean.

Original entry on oeis.org

1, 2, 0, 8, 2, 68, 32, 720, 464, 8480, 6656, 106368, 95912, 1390928, 1392512, 18734144, 20371650, 257955716, 300101760, 3613109008, 4448177412, 51302395528, 66289160512, 736588435360, 992578330048, 10674012880512, 14924667774976, 155890890782720, 225244659392784, 2291995151532576, 3410654921389824
Offset: 0

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Author

Joerg Arndt, Aug 14 2013

Keywords

Comments

Convolution square is A092266.

Links

Crossrefs

Cf. A092266 (1+4*x)/AGM(1+4*x,1-4*x).
Cf. A081085 1/AGM(1,1-8*x), A053175 1/AGM(1,1-16*x), A090004 1/AGM(1,1-16*x)^(1/2), A089602 1/AGM(1,1-16*x)^(1/4).

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[2*(1 + 4*x)*EllipticK[1 - (1 + 4*x)^2/(1 - 4*x)^2] / (Pi*(1 - 4*x))], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 27 2019 *)
  • PARI
    Vec( 1/agm(1,(1-4*x)/(1+4*x)+O(x^66))^(1/2) ) \\ Joerg Arndt, Aug 14 2013

Formula

a(n) ~ 2^(2*n - 1/2) / (n*sqrt(Pi*log(n))) * (1 - (gamma + 3*log(2)) / (2*log(n)) + (3*gamma^2/8 + 9*gamma*log(2)/4 + 27*log(2)^2/8 - 1/16*Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

A250029 Maximum number of binary strings with symmetrically partitioned n 1's and n 0's, counted up to isomorphism.

Original entry on oeis.org

1, 1, 1, 4, 9, 16, 36, 144, 400, 900, 3600, 11025, 28224, 78400, 254016, 705600, 2286144, 6350400, 25401600, 85377600, 250905600, 768398400, 3073593600, 10600761600, 32464832400, 129859329600, 456536705625
Offset: 0

Views

Author

Andrei Cretu, Nov 11 2014

Keywords

Comments

The number of binary strings, counted up to isomorphism, that can be constructed by taking an equal number (n) of 0's and 1's and partitioning both the 0's and the 1's into m runs using the same partition, can be written as:
dualseq[partition[n]]=m!^2/(Prod_j(m_j!))^2
where m_j is the multiplicity of runs of length j.
The numbers satisfy the relations Sum_j(m_j)=m, Sum_j(j*m_j)=n.
The strings obtained in this manner are a subset of those in A247651.
Both the finest and coarsest partitions of n minimize dualseq[partition[n]]. In this sense, dualseq[partition[n]] is another relative measure of the complexity of the partition and the associated binary strings.
a[n] is the number of strings, counted up to isomorphism, that can be generated based on the partition that maximizes dualseq[partition[n]].

Examples

			n=0 gives the empty string.
n=1 and the only possible partition generate 01 (and the isomorphic 10).
For n=2, both possible partitions generate, up to isomorphism, 1 string, 0011 (1100), and respectively 0101 (1010).
For n=3, the optimal partition is {1,2}, generating, up to isomorphism, 4 strings: 001011 (110100), 001101 (110010), 010011 (101100) and 011001 (100110).
For n=4, the optimal partition is {1,1,2}, generating, up to isomorphism, 9 strings: 00101011 (11010100), 00101101 (11010010), 00110101 (11001010), 01001011 (10110100), 01001101 (10110010), 01010011 (10101100), 01011001 (10100110), 01100101 (10011010) and 01101001 (10010110).

Crossrefs

Cf. A247651, A046952, A092266, A136404, A176471, A065886.

Programs

  • Mathematica
    dualseq[p_]:=Factorial[Length[p]]^2/Apply[Times,Map[Factorial[Count[p,#1]]&,Range[Max[Length[p]]]]]^2
a[n_]:=Max[Map[dualseq,IntegerPartitions[n]]]
Table[a[n],{n,0,25}] (* after A130670 *)

Formula

a[n]=Max[m!^2/(Prod_j(m_j!))^2] where Sum_j(m_j)=m, Sum_j(j*m_j)=n, over all partitions of n.
Showing 1-2 of 2 results.

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