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.

A339659 Irregular triangle read by rows where T(n,k) is the number of graphical partitions of 2n into k parts, 0 <= k <= 2n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 2, 3, 2, 1, 1, 0, 0, 0, 0, 1, 4, 5, 3, 2, 1, 1, 0, 0, 0, 0, 1, 4, 7, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 4, 9, 11, 11, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 2, 11, 15, 17, 15, 11, 7, 5, 3, 2, 1, 1
Offset: 0

Views

Author

Gus Wiseman, Dec 18 2020

Keywords

Comments

Conjecture: The column sums 1, 0, 1, 2, 7, 20, 67, ... are given by A304787.
An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. Graphical partitions are counted by A000569.

Examples

			Triangle begins:
  1
  0 0 1
  0 0 0 1 1
  0 0 0 1 2 1 1
  0 0 0 0 2 3 2 1 1
  0 0 0 0 1 4 5 3 2 1 1
  0 0 0 0 1 4 7 7 5 3 2 1 1
For example, row n = 5 counts the following partitions:
  3322  22222  222211  2221111  22111111  211111111  1111111111
        32221  322111  3211111  31111111
        33211  331111  4111111
        42211  421111
               511111

Crossrefs

A000569 gives the row sums.
A004250 is the central column.
A005408 gives the row lengths.
A008284/A072233 is the version counting all partitions.
A259873 is the left half of the triangle.
A309356 is a universal embedding.
A027187 counts partitions of even length.
A339559 = partitions that cannot be partitioned into distinct strict pairs.
A339560 = partitions that can be partitioned into distinct strict pairs.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A000569 counts graphical partitions (A320922).
- A058696 counts partitions of 2n (A300061).
- A147878 counts connected multigraphical partitions (A320925).
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
- A339617 counts non-graphical partitions of 2n (A339618).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
Cf. A000219, A002100, A006881, A007717, A025065, A320656, A320894, A338914, A338916, A339561, A339661.

Programs

  • Mathematica
    prpts[m_]:=If[Length[m]==0,{{}},Join@@Table[Prepend[#,ipr]&/@prpts[Fold[DeleteCases[#1,#2,{1},1]&,m,ipr]],{ipr,Subsets[Union[m],{2}]}]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Length[Union[#]]==k&&Select[prpts[#],UnsameQ@@#&]!={}&]],{n,0,5},{k,0,2*n}]

A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.

Original entry on oeis.org

1, 1, 3, 8, 25, 77, 256, 854, 2940, 10229, 36124, 128745, 463137, 1677816, 6118165, 22432778, 82660369, 305916561, 1136621136, 4238006039, 15852603939, 59471304434, 223704813807, 843547443903, 3188064830876, 12074092672950, 45816941923597, 174173975322767
Offset: 0

Views

Author

N. J. A. Sloane, Nov 06 2003

Keywords

Comments

a(n) is the number of forests of rooted plane binary trees (each node has outdegree = 0 or 2) where the trees have a total of n internal nodes. Cf. A222006. - Geoffrey Critzer, Feb 26 2013

Links

Crossrefs

Row sums of A275431.
Cf. A222006, A304787.

Programs

  • Magma
    m:=35;
f:= func< x | (&*[1/(1-x^j)^Catalan(j): j in [1..m+2]]) >;
R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( f(x) )); // G. C. Greubel, Dec 12 2022
  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
       binomial(2*d, d)/(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2012
  • Mathematica
    With[{nn=35}, CoefficientList[Series[Product[1/(1-x^i)^CatalanNumber[i], {i,nn}], {x,0,nn}], x]] (* Geoffrey Critzer, Feb 26 2013 *)
  • SageMath
    # uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(2*n, n)/(n+1))
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

Formula

Euler transform of Catalan numbers (A000108). - Franklin T. Adams-Watters, Mar 01 2006
a(n) ~ c * 4^n / n^(3/2), where c = exp(Sum_{k>=1} (-2 + 4^k - 4^k*sqrt(1 - 4^(1-k)))/(2*k) ) / sqrt(Pi) = 1.60022306097485382475864802335610662545... - Vaclav Kotesovec, Mar 21 2021

A344130 Expansion of Product_{k>=1} (1 + x^k)^binomial(2*k,k).

Original entry on oeis.org

1, 2, 7, 32, 131, 562, 2383, 10124, 42916, 181844, 769246, 3250388, 13716377, 57812466, 243382957, 1023463628, 4299199426, 18040918456, 75632083258, 316774424568, 1325591467994, 5542462776932, 23155074355078, 96661979245880, 403223735948096, 1680858909265768
Offset: 0

Views

Author

Vaclav Kotesovec, May 10 2021

Keywords

Crossrefs

Cf. A000984, A304787, A344108.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[(1 + x^k)^Binomial[2 k, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[Exp[Sum[-(-1)^j * (1/Sqrt[1 - 4*x^j] - 1)/j, {j, 1, nmax}]], {x, 0, nmax}], x]

Formula

a(n) ~ 2^(2*n - 1/3) * exp(3*n^(1/3)/2^(2/3) - 1 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} (-1)^k * (1 - 1/sqrt(1 - 4^(1-k)))/k = -0.0680700790487788003241709878640131435678833005035785286095068...
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.