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.

Previous Showing 31-39 of 39 results.

A358732 Number of labeled trees covering 2n nodes, half of which are leaves.

Original entry on oeis.org

0, 12, 720, 109200, 31752000, 15186346560, 10852244282880, 10851787634688000, 14481281691676800000, 24881574582258352358400, 53525038934303849706393600, 140958354488116955062668595200, 446153762528143389466306560000000, 1671353230826683972965623004979200000
Offset: 1

Views

Author

Gus Wiseman, Dec 01 2022

Keywords

Examples

			The a(2) = 12 trees:
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,3},{3,4}}
  {{1,2},{1,4},{2,3}}
  {{1,2},{1,4},{3,4}}
  {{1,2},{2,3},{3,4}}
  {{1,2},{2,4},{3,4}}
  {{1,3},{1,4},{2,3}}
  {{1,3},{1,4},{2,4}}
  {{1,3},{2,3},{2,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4}}
  {{1,4},{2,3},{3,4}}

Links

Crossrefs

A central column of A055314.
The unlabeled rooted version is A185650.
The unlabeled version is A358107.
A000272 counts trees, bisection A163395.
A001187 counts connected graphs.
A006129 counts covering graphs.
A014068 counts graphs with n vertices and n-1 edges.
Cf. A000055, A001349, A006125.

Programs

  • Mathematica
    a[n_]:=StirlingS2[2*n-2, n]*(2*n)!/n!; Array[a,14] (* Stefano Spezia, Aug 02 2024 *)
  • PARI
    a(n) = stirling(2*n-2, n, 2)*(2*n)!/n! \\ Andrew Howroyd, Dec 30 2022

Formula

a(n) = A055314(2*n, n) = Stirling2(2*n-2, n)*(2*n)!/n!. - Andrew Howroyd, Dec 30 2022

Extensions

Terms a(6) and beyond from Andrew Howroyd, Dec 30 2022

A359398 Number of unlabeled trees covering 2n nodes, half of which are leaves.

Original entry on oeis.org

0, 1, 2, 8, 32, 158, 833, 4755, 28389, 176542, 1131055, 7432876, 49873477, 340658595, 2362652648, 16605707901, 118082160358, 848399575321, 6152038125538, 44981009272740, 331344933928536, 2457372361637286, 18337490246234464, 137612955519565773, 1038076541372187991
Offset: 1

Views

Author

Gus Wiseman, Jan 01 2023

Keywords

Links

Crossrefs

Left of central column of A055290.
The labeled version is the left of central column of A055314.
The rooted version is A185650.
For n+1 leaves we have A358107.
The labeled version is A358732.
A000272 counts trees, bisection A163395, unlabeled A000055.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts graphs with n vertices and n-1 edges, unlabeled A001433.

Formula

a(n) = A055290(2*n, n). - Andrew Howroyd, Jan 01 2023

Extensions

Terms a(12) and beyond from Andrew Howroyd, Jan 01 2023

A298690 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1)/2, n) * x^n / A(x)^( n*(n+1)/2 ).

Original entry on oeis.org

1, 1, 2, 10, 83, 971, 14679, 271065, 5887674, 146573343, 4106195739, 127709962780, 4364136955874, 162503129082497, 6548680061635319, 283973223632787150, 13185195626147207058, 652695122347799336199, 34316223642036784123819, 1909798106976656110119169, 112165977515060359849066878, 6933265352057611483132200642
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 83*x^4 + 971*x^5 + 14679*x^6 + 271065*x^7 + 5887674*x^8 + 146573343*x^9 + 4106195739*x^10 + 127709962780*x^11 + 4364136955874*x^12 + 162503129082497*x^13 + 6548680061635319*x^14 + 283973223632787150*x^15 + ...
such that
A(x) = 1 + C(1,1)*x/A(x) + C(3,2)*x^2/A(x)^3 + C(6,3)*x^3/A(x)^6 + C(10,4)*x^4/A(x)^10 + C(15,5)*x^5/A(x)^15 + C(21,6)*x^6/A(x)^21 + C(28,7)*x^7/A(x)^28 + ...
more explicitly,
A(x) = 1 + x/A(x) + 3*x^2/A(x)^3 + 20*x^3/A(x)^6 + 210*x^4/A(x)^10 + 3003*x^5/A(x)^15 + 54264*x^6/A(x)^21 + 1184040*x^7/A(x)^28 + 30260340*x^8/A(x)^36 + ...

Crossrefs

Cf. A014068, A298689, A298691.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1)/2,m) * x^m/Ser(A)^(m*(m+1)/2) ))); G=Ser(A); A[n+1]}
for(n=0,30,print1(a(n),", "))

A137345 a(n) = binomial( n(n+1)/2, n) mod n.

Original entry on oeis.org

0, 1, 2, 2, 3, 0, 4, 4, 5, 0, 6, 7, 7, 0, 9, 8, 9, 0, 10, 0, 3, 0, 12, 6, 13, 13, 5, 0, 15, 4, 16, 16, 24, 0, 20, 12, 19, 0, 4, 38, 21, 0, 22, 22, 30, 0, 24, 12, 25, 25, 18, 0, 27, 0, 45, 51, 51, 0, 30, 0, 31, 0, 35, 32, 35, 0, 34, 0, 33, 10, 36, 0, 37, 0, 39, 57, 14, 0, 40, 12, 5, 0, 42, 40
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Apr 08 2008, Apr 14 2008

Keywords

Crossrefs

Cf. A139125.

Programs

  • Maple
    P:=proc(n) local a,b,i; a:=0; for i from 1 by 1 to n do a:=a+i; b:=(binomial(a,i) mod i); print(b); od; end: P(101);

Formula

a(n) = A014068(n) mod n. - R. J. Mathar, Apr 09 2008

A189831 Labeled simple graphs with n nodes and n-1 edges that are not trees.

Original entry on oeis.org

0, 0, 0, 4, 85, 1707, 37457, 921896, 25477371, 786163135, 26890701739, 1012165431744, 41638805754078, 1860589088529164, 89802422444553825, 4658465562594667088, 258566755450911870007, 15294477441385413149679, 960641026388207044487891, 63861339527473864490450300
Offset: 1

Views

Author

Geoffrey Critzer, Apr 28 2011

Keywords

Comments

Equivalently a(n) is the number of labeled simple graphs on n nodes having n-1 edges that have at least two connected components.
Evidently almost all such graphs are disconnected.

Examples

			a(4) = 4 because there are 20 labeled simple graphs on four nodes with three edges but 16 of these are connected i.e. they are trees.

Links

Crossrefs

Cf. A084546, A000272, A014068.

Programs

  • Magma
    [Binomial(Binomial(n,2),n-1) - n^(n-2): n in [1..20]]; // G. C. Greubel, Jan 14 2018
  • Mathematica
    Table[Binomial[Binomial[n,2],n-1]-n^(n-2),{n,1,20}]
  • PARI
    for(n=1,20, print1(binomial(binomial(n,2),n-1) - n^(n-2), ", ")) \\ G. C. Greubel, Jan 14 2018

Formula

a(n) = C(C(n,2),n-1) - n^(n-2) = A014068(n-1)-A000272(n), where C(x,y) is the binomial coefficient.

A304403 G.f. A(x) satisfies: [x^n] A(x) * (1+x)^(n*(n-1)/2) = [x^n] (1+x)^(n*(n+1)/2) for n >= 0.

Original entry on oeis.org

1, 1, 2, 10, 85, 1001, 15036, 273932, 5858560, 143735650, 3976623010, 122427035732, 4150101179665, 153570442058684, 6158890134498661, 266074435570524219, 12318188650382356297, 608375312208623381681, 31927089332971578025902, 1774159611825531210120848, 104068978022940760659203857, 6425920321260029677988702979, 416624218261246444665784515673
Offset: 0

Views

Author

Paul D. Hanna, May 18 2018

Keywords

Comments

Limit ( a(n) / n! )^(1/n) seems to exist and is near 3.1...

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 85*x^4 + 1001*x^5 + 15036*x^6 + 273932*x^7 + 5858560*x^8 + 143735650*x^9 + 3976623010*x^10 + 122427035732*x^11 + ...
such that
binomial(n*(n+1)/2, n) = Sum_{k=0..n} a(k) * binomial(n*(n-1)/2, n-k) for n >= 0.
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x) * (1+x)^(n*(n-1)/2) begins:
  n=0: [1, 1, 2, 10, 85, 1001, 15036, 273932, 5858560, ...];
  n=1: [1, 1, 2, 10, 85, 1001, 15036, 273932, 5858560, ...];
  n=2: [1, 2, 3, 12, 95, 1086, 16037, 288968, 6132492, ...];
  n=3: [1, 4, 8, 20, 122, 1288, 18304, 322128, 6726465, ...];
  n=4: [1, 7, 23, 57, 210, 1722, 22554, 381026, 7749049, ...];
  n=5: [1, 11, 57, 195, 605, 3003, 30953, 482471, 9415575, ...];
  n=6: [1, 16, 122, 600, 2265, 8604, 54264, 674348, 12170710, ...];
  n=7: [1, 22, 233, 1592, 8030, 33880, 153790, 1184040, 17240535, ...];
  n=8: [1, 29, 408, 3720, 24872, 132468, 623924, 3313868, 30260340, ...]; ...
in which the main diagonal equals A014068:
[1, 1, 3, 20, 210, 3003, 54264, 1184040, ..., binomial(n*(n+1)/2, n), ...]
illustrating [x^n] A(x) * (1+x)^(n*(n-1)/2) = [x^n] (1+x)^(n*(n+1)/2) for n >= 0.
ILLUSTRATION OF RECURRENCE.
The table of coefficients of x^k in (1+x)^(n*(n-1)/2) begins:
  n=1: [1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
  n=2: [1, 1, 0, 0, 0, 0, 0, 0, 0, ...];
  n=3: [1, 3, 3, 1, 0, 0, 0, 0, 0, ...];
  n=4: [1, 6, 15, 20, 15, 6, 1, 0, 0, ...];
  n=5: [1, 10, 45, 120, 210, 252, 210, 120, 45, ...];
  n=6: [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, ...];
  n=7: [1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, ...];
  n=8: [1, 28, 378, 3276, 20475, 98280, 376740, 1184040, 3108105, ...];
  n=9: [1, 36, 630, 7140, 58905, 376992, 1947792, 8347680, 30260340, ...]; ...
The recurrence uses the above coefficients like so:
  a(0) = 1;
  a(1) = 1 - (0*a(0)) = 1;
  a(2) = 3 - (0*a(0) + 1*a(1)) = 2;
  a(3) = 20 - (1*a(0) + 3*a(1) + 3*a(2)) = 10;
  a(4) = 210 - (15*a(0) + 20*a(1) + 15*a(2) + 6*a(3)) = 85;
  a(5) = 3003 - (252*a(0) + 210*a(1) + 120*a(2) + 45*a(3) + 10*a(4)) = 1001; ...
illustrating a(n) = C(n*(n+1)/2, n) - Sum_{k=0..n-1} C(n*(n-1)/2, n-k) * a(k), for n >= 0.

Links

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A-1; A[#A] = binomial(m*(m+1)/2, m) - polcoeff( Ser(A)*(1+x +x*O(x^m))^(m*(m-1)/2) , m) );A[n+1]}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* Recurrence: */
{a(n) = if(n==0,1, binomial(n*(n+1)/2, n) - sum(k=0,n-1, a(k) * binomial(n*(n-1)/2, n-k) ) )}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = binomial(n*(n+1)/2, n) - Sum_{k=0..n-1} a(k) * binomial(n*(n-1)/2, n-k), for n >= 0.

A369198 Number of labeled loop-graphs with n vertices and the same number of edges as covered vertices.

Original entry on oeis.org

1, 2, 6, 30, 241, 2759, 40824, 736342, 15622835, 380668095, 10467815086, 320529284621, 10813165015074, 398413594789777, 15917197015926392, 685312404706694574, 31631317971844128229, 1558017329350990780607, 81567807853701988869120, 4522975947689168088308305
Offset: 0

Views

Author

Gus Wiseman, Jan 18 2024

Keywords

Examples

			The a(0) = 1 through a(3) = 30 loop-graphs (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1},{2}}    {{3}}
             {{1},{1,2}}  {{1},{2}}
             {{2},{1,2}}  {{1},{3}}
                          {{2},{3}}
                          {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

Crossrefs

The version counting all vertices is A014068.
The loopless case is A367862, counting all vertices A116508.
The covering case is A368597, connected A368951.
With inequality we have A369196, covering A369194, connected A369197.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000272, A000666, A006649, A062740, A066383, A367863, A369192, A369193.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[#]==Length[Union@@#]&]],{n,0,5}]

Formula

Binomial transform of A368597.

A386879 a(n) = [x^n] 1/(1 - x)^(n*(n-1)/2).

Original entry on oeis.org

1, 0, 1, 10, 126, 2002, 38760, 888030, 23535820, 708930508, 23930713170, 895068996640, 36749279048405, 1643385429346680, 79515468511191440, 4139207762053520646, 230672804560960311000, 13703037308872895467960, 864424422377992704918690, 57711135174726478041405270, 4065392394346039279040037520
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 06 2025

Keywords

Crossrefs

Cf. A014068, A054688, A116508, A177234, A386880.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1-x)^(n*(n-1)/2), {x, 0, n}], {n, 0, 25}]
Join[{1}, Table[Binomial[n*(n+1)/2, n] * (n-1) / (n+1), {n, 1, 25}]]

Formula

a(n) ~ exp(n) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)).
For n > 0, a(n) = binomial(n*(n+1)/2, n) * (n-1)/(n+1).

A386880 a(n) = [x^n] 1/(1 - x)^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 6, 56, 715, 11628, 230230, 5379616, 145008513, 4431613550, 151473214816, 5727160371180, 237377895350076, 10704005376506540, 521748877569771510, 27338999059076777600, 1532576541123942256285, 91527291781199227579626, 5801648509628587739612170, 389032765009190361630625600
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 06 2025

Keywords

Crossrefs

Cf. A014068, A054688, A116508, A177234, A386879.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1-x)^(n*(n+1)/2), {x, 0, n}], {n, 0, 25}]
Join[{1}, Table[Binomial[n*(n + 3)/2, n]*(n + 1)/(n + 3), {n, 1, 25}]]

Formula

a(n) ~ exp(n+2) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)).
For n > 0, a(n) = binomial(n*(n+3)/2, n) * (n+1)/(n+3).
Previous Showing 31-39 of 39 results.

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