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 1-9 of 9 results.

A330942 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 7, 1, 1, 1, 8, 75, 32, 1, 1, 1, 16, 1105, 2712, 161, 1, 1, 1, 32, 20821, 449102, 116681, 842, 1, 1, 1, 64, 478439, 122886128, 231522891, 5366384, 4495, 1, 1, 1, 128, 12977815, 50225389432, 975712562347, 131163390878, 256461703, 24320, 1, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 13 2020

Keywords

Comments

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
A(n,k) is the number of labeled n-uniform hypergraphs with multiple edges allowed and with k edges and no isolated vertices. When n=2 these objects are multigraphs.

Examples

			Array begins:
============================================================
n\k | 0 1    2         3              4                5
----+-------------------------------------------------------
  0 | 1 1    1         1              1                1 ...
  1 | 1 1    2         4              8               16 ...
  2 | 1 1    7        75           1105            20821 ...
  3 | 1 1   32      2712         449102        122886128 ...
  4 | 1 1  161    116681      231522891     975712562347 ...
  5 | 1 1  842   5366384   131163390878 8756434117294432 ...
  6 | 1 1 4495 256461703 78650129124911 ...
  ...
The A(2,2) = 7 matrices are:
   [1 0]  [1 0]  [1 0]  [1 1]  [1 0]  [1 0]  [1 1]
   [1 0]  [0 1]  [0 1]  [1 0]  [1 1]  [0 1]  [1 1]
   [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [1 1]
   [0 1]  [0 1]  [1 0]

Links

Crossrefs

Rows n=1..3 are A000012, A121316, A136246.
Columns k=0..3 are A000012, A000012, A226994, A137220.
The version with nonnegative integer entries is A331315.
Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:
All solutions: A262809 (all), A331567 (distinct rows).
Up to row permutation: A188392, A188445, A331126, A331039.
Up to column permutation: this sequence, A331571, A331277, A331569.
Nonisomorphic: A331461, A331510, A331508, A331509.
Cf. A331638.

Programs

  • Mathematica
    T[n_, k_] := With[{m = n k}, Sum[Binomial[Binomial[j, n] + k - 1, k] Sum[ (-1)^(i - j) Binomial[i, j], {i, j, m}], {j, 0, m}]];
Table[T[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 10 2020, from PARI *)
  • PARI
    T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j, n)+k-1, k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}

Formula

A(n,k) = Sum_{j=0..n*k} binomial(binomial(j,n)+k-1, k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A262809(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331277(n, j).
A331638(n) = Sum_{d|n} A(n/d, d).

A014500 Number of graphs with unlabeled (non-isolated) nodes and n labeled edges.

Original entry on oeis.org

1, 1, 2, 9, 70, 794, 12055, 233238, 5556725, 158931613, 5350854707, 208746406117, 9315261027289, 470405726166241, 26636882237942128, 1678097862705130667, 116818375064650241036, 8932347052564257212796, 746244486452472386213939, 67796741482683128375533560
Offset: 0

Views

Author

Simon Plouffe, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)

Keywords

References

  • G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Links

Crossrefs

Row n=2 of A331126.
Cf. A020554, A020555, A014501, A060053.

Programs

  • Maple
    read("transforms") ;
A020556 := proc(n) local k; add((-1)^(n+k)*binomial(n, k)*combinat[bell](n+k), k=0..n) end proc:
A014500 := proc(n) local i,gexp,lexp;
gexp := [seq(1/2^i/i!,i=0..n+1)] ;
lexp := add( A020556(i)*((log(1+x))/2)^i/i!,i=0..n+1) ;
lexp := taylor(lexp,x=0,n+1) ;
lexp := gfun[seriestolist](lexp,'ogf') ;
CONV(gexp,lexp) ; op(n+1,%)*n! ; end proc:
seq(A014500(n),n=0..20) ; # R. J. Mathar, Jul 03 2011
  • Mathematica
    max = 20; A020556[n_] := Sum[(-1)^(n+k)*Binomial[n, k]*BellB[n+k], {k, 0, n}]; egf = Exp[x/2]*Sum[A020556[n]*(Log[1+x]/2)^n/n!, {n, 0, max}] + O[x]^max; CoefficientList[egf, x]*Range[0, max-1]! (* Jean-François Alcover, Feb 19 2017, after Vladeta Jovovic *)
  • PARI
    \\ here egf1 is A020556 as e.g.f.
egf1(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i,k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
seq(n)={my(B=egf1(n), L=log(1+x + O(x*x^n))/2); Vec(serlaplace(exp(x/2 + O(x*x^n))*sum(k=0, n, polcoef(B ,k)*L^k)))} \\ Andrew Howroyd, Jan 13 2020

Formula

E.g.f.: exp(-1+x/2)*Sum((1+x)^binomial(n, 2)/n!, n=0..infinity) [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004
E.g.f.: exp(x/2)*Sum(A020556(n)*(log(1+x)/2)^n/n!, n=0..infinity). - Vladeta Jovovic, May 02 2004
Binomial transform of A060053.
The e.g.f.'s of A020554 (S(x)) and A014500 (U(x)) are related by S(x) = U(e^x-1).
The e.g.f.'s of A014500 (U(x)) and A060053 (V(x)) are related by U(x) = e^x*V(x).

A331039 Array read by antidiagonals: A(n,k) is the number of T_0 n-regular set-systems on a k-set.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 5, 0, 0, 1, 0, 1, 43, 5, 0, 0, 1, 0, 1, 518, 175, 1, 0, 0, 1, 0, 1, 8186, 9426, 272, 0, 0, 0, 1, 0, 1, 163356, 751365, 64453, 205, 0, 0, 0, 1, 0, 1, 3988342, 84012191, 23553340, 248685, 80, 0, 0, 0, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 08 2020

Keywords

Comments

An n-regular set-system is a finite set of nonempty sets in which each element appears in n blocks.
A set-system is T_0 if for every two distinct elements there exists a block containing one but not the other element.
A(n,k) is the number of binary matrices with k distinct columns and any number of distinct nonzero rows with n ones in every column and rows in decreasing lexicographic order.

Examples

			Array begins:
==========================================================
n\k | 0 1 2 3   4       5           6                7
----+-----------------------------------------------------
  0 | 1 1 0 0   0       0           0                0 ...
  1 | 1 1 1 1   1       1           1                1 ...
  2 | 1 0 1 5  43     518        8186           163356 ...
  3 | 1 0 0 5 175    9426      751365         84012191 ...
  4 | 1 0 0 1 272   64453    23553340      13241130441 ...
  5 | 1 0 0 0 205  248685   421934358    1176014951129 ...
  6 | 1 0 0 0  80  620548  5055634889   69754280936418 ...
  7 | 1 0 0 0  15 1057989 43402628681 2972156676325398 ...
  ...
The A(2,3) = 5 matrices are:
  [1 1 1]    [1 1 0]    [1 1 0]    [1 0 1]    [1 1 0]
  [1 0 0]    [1 0 1]    [1 0 0]    [1 0 0]    [1 0 1]
  [0 1 0]    [0 1 0]    [0 1 1]    [0 1 1]    [0 1 1]
  [0 0 1]    [0 0 1]    [0 0 1]    [0 1 0]
The corresponding set-systems are:
  {{1,2,3}, {1}, {2}, {3}},
  {{1,2}, {1,3}, {2,3}},
  {{1,2}, {1,3}, {2}, {3}},
  {{1,2}, {1}, {2,3}, {3}},
  {{1,3}, {1}, {2,3}, {2}}.

Links

Crossrefs

Rows n=1..4 are A000012, A060053, A060070, A331655.
Cf. A188445, A330942, A330964, A331126, A331160, A331161, A331569, A331654.

Programs

  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}

Formula

A(n, k) = Sum_{j=1..k} Stirling1(k, j)*A188445(n, j) for n, k >= 1.
A(n, k) = 0 for k >= 1, n > 2^(k-1).
A331654(n) = Sum_{d|n} A(n/d, d).

A331508 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set multipartitions (multisets of sets) on a k-set.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 3, 1, 1, 0, 1, 11, 12, 4, 1, 1, 0, 1, 26, 66, 25, 5, 1, 1, 0, 1, 68, 445, 278, 44, 6, 1, 1, 0, 1, 177, 4279, 5532, 966, 73, 7, 1, 1, 0, 1, 497, 53340, 200589, 53535, 2957, 112, 8, 1, 1, 0, 1, 1476, 846254, 11662671, 7043925, 431805, 8149, 166, 9, 1, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 18 2020

Keywords

Comments

An n-regular set multipartition is a finite multiset of nonempty sets in which each element appears in n blocks.
A set multipartition is T_0 if for every two distinct elements there exists a block containing one but not the other element.
A(n,k) is the number of nonequivalent binary matrices with k distinct columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.
A(n,k) is the number of non-isomorphic set-systems with k parts each of size n.

Examples

			Array begins:
===============================================
n\k | 0 1 2  3    4      5       6        7
----+------------------------------------------
  0 | 1 1 0  0    0      0       0        0 ...
  1 | 1 1 1  1    1      1       1        1 ...
  2 | 1 1 2  5   11     26      68      177 ...
  3 | 1 1 3 12   66    445    4279    53340 ...
  4 | 1 1 4 25  278   5532  200589 11662671 ...
  5 | 1 1 5 44  966  53535 7043925 ...
  6 | 1 1 6 73 2957 431805 ...
  ...
The A(2,3) = 5 matrices are:
  [1 0 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 0]
  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 1]  [1 0 1]
  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 1]
  [0 1 0]  [0 0 1]  [0 0 1]  [0 0 1]
  [0 0 1]  [0 0 1]
  [0 0 1]

Links

Crossrefs

Rows n=1..6 are A000012, A000664, A058790, A058791, A058792, A058793.
Cf. A283877, A306019, A330942, A331126, A331277, A331461, A331509, A331510.

Programs

  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))[k]}
T(n,k)={my(m=n*k, s=0); if(m==0, k<=1, forpart(q=m, my(g=sum(t=1, k, K(q, t, n)*x^t/t) + O(x*x^k)); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), k)); s/m!)}
{ for(n=0, 6, for(k=0, 5, print1(T(n, k), ", ")); print) } \\ Andrew Howroyd, Jan 16 2024

Formula

A306019(n) = Sum_{d|n} A(n/d, d).

A331160 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of distinct nonzero rows with column sums n and rows in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 2, 1, 0, 1, 27, 15, 2, 1, 0, 1, 266, 317, 44, 3, 1, 0, 1, 3599, 12586, 2763, 120, 4, 1, 0, 1, 62941, 803764, 390399, 21006, 319, 5, 1, 0, 1, 1372117, 75603729, 103678954, 10074052, 147296, 804, 6, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 10 2020

Keywords

Comments

The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows up to permutation of rows.

Examples

			Array begins:
===================================================================
n\k | 0 1   2      3          4              5                6
----+--------------------------------------------------------------
  0 | 1 1   0      0          0              0                0 ...
  1 | 1 1   1      1          1              1                1 ...
  2 | 1 1   4     27        266           3599            62941 ...
  3 | 1 2  15    317      12586         803764         75603729 ...
  4 | 1 2  44   2763     390399      103678954      46278915417 ...
  5 | 1 3 120  21006   10074052    10679934500   21806685647346 ...
  6 | 1 4 319 147296  232165926   956594630508 8717423133548684 ...
  7 | 1 5 804 967829 4903530137 76812482919237 ...
      ...
The A(2,2) = 4 matrices are:
   [2 1]   [2 0]   [1 2]   [1 1]
   [0 1]   [0 2]   [1 0]   [1 0]
                           [0 1]

Links

Crossrefs

Rows n=1..3 are A000012, A331316, A331344
Columns k=0..2 are A000012, A000009, A331317.
Cf. A219585, A331039, A331126, A331161, A331315, A331318.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}

Formula

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A219585(n, j).
A331318(n) = Sum_{d|n} A(n/d, d).

A331161 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and rows in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 7, 3, 1, 0, 1, 43, 28, 5, 1, 0, 1, 403, 599, 104, 7, 1, 0, 1, 5245, 23243, 6404, 332, 11, 1, 0, 1, 89132, 1440532, 872681, 57613, 1032, 15, 1, 0, 1, 1898630, 131530132, 222686668, 26560747, 473674, 2983, 22, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 10 2020

Keywords

Comments

The condition that the rows be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of rows

Examples

			Array begins:
====================================================================
n\k | 0  1    2       3           4             5              6
----+---------------------------------------------------------------
  0 | 1  1    0       0           0             0              0 ...
  1 | 1  1    1       1           1             1              1 ...
  2 | 1  2    7      43         403          5245          89132 ...
  3 | 1  3   28     599       23243       1440532      131530132 ...
  4 | 1  5  104    6404      872681     222686668    95605470805 ...
  5 | 1  7  332   57613    26560747   26852940027 52296207431182 ...
  6 | 1 11 1032  473674   712725249 2776638423133 ...
  7 | 1 15 2983 3599384 17328777789 ...
  ...
The A(2,2) = 7 matrices are:
   [2 1]   [2 0]   [1 2]   [1 1]   [2 0]   [1 0]   [1 0]
   [0 1]   [0 2]   [1 0]   [1 0]   [0 1]   [1 0]   [1 0]
                           [0 1]   [0 1]   [0 2]   [0 1]
                                                   [0 1]

Links

Crossrefs

Rows n=1..3 are A000012, A014501, A331196.
Columns k=0..2 are A000012, A000041, A331197.
Cf. A219727, A330158, A331039, A331126, A331160, A331315.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, k<=1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p, n, k), [1, n]); s*q[#q-j]))}

Formula

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A219727(n, j).
A330158(n) = Sum_{d|n} A(n/d, d).

A331389 Number of binary matrices with n distinct columns and any number of nonzero rows with 3 ones in every column and rows in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 3, 29, 666, 28344, 1935054, 193926796, 26892165502, 4946464286746, 1168900475263013, 346080409272270888, 125798338606148948325, 55204084562033205121607, 28834556615453989801860765, 17710828268156331289770544579, 12658784968736373972502731143309
Offset: 0

Views

Author

Andrew Howroyd, Jan 15 2020

Keywords

Comments

The condition that the rows be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of rows.
a(n) is the number of T_0 3-regular set multipartitions (multisets of sets) on an n-set.

Examples

			The a(2) = 3 matrices are:
   [1 0]   [1 1]   [1 1]
   [1 0]   [1 0]   [1 1]
   [1 0]   [1 0]   [1 0]
   [0 1]   [0 1]   [0 1]
   [0 1]   [0 1]
   [0 1]

Links

Crossrefs

Row n=3 of A331126.
Cf. A165434.

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A165434(k). - Andrew Howroyd, Jan 31 2020

A331390 Number of binary matrices with 3 distinct columns and any number of nonzero rows with n ones in every column and rows in nonincreasing lexicographic order.

Original entry on oeis.org

1, 9, 29, 68, 134, 237, 388, 600, 887, 1265, 1751, 2364, 3124, 4053, 5174, 6512, 8093, 9945, 12097, 14580, 17426, 20669, 24344, 28488, 33139, 38337, 44123, 50540, 57632, 65445, 74026, 83424, 93689, 104873, 117029, 130212, 144478, 159885, 176492, 194360, 213551
Offset: 1

Views

Author

Andrew Howroyd, Jan 15 2020

Keywords

Comments

The condition that the rows be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of rows.
a(n) is the number of T_0 n-regular set multipartitions (multisets of sets) on a 3-set.

Examples

			The a(2) = 9 matrices are:
[1, 0, 0]  [1, 1, 0]  [1, 0, 1]  [1, 0, 0]
[1, 0, 0]  [1, 0, 0]  [1, 0, 0]  [1, 0, 0]
[0, 1, 0]  [0, 1, 0]  [0, 1, 0]  [0, 1, 1]
[0, 1, 0]  [0, 0, 1]  [0, 1, 0]  [0, 1, 0]
[0, 0, 1]  [0, 0, 1]  [0, 0, 1]  [0, 0, 1]
[0, 0, 1]
.
[1, 1, 1]  [1, 1, 0]  [1, 1, 0]  [1, 0, 1]  [1, 1, 0]
[1, 0, 0]  [1, 0, 1]  [1, 0, 0]  [1, 0, 0]  [1, 0, 1]
[0, 1, 0]  [0, 1, 0]  [0, 1, 1]  [0, 1, 1]  [0, 1, 1]
[0, 0, 1]  [0, 0, 1]  [0, 0, 1]  [0, 1, 0]

Links

Crossrefs

Column k=3 of A331126.

Programs

  • PARI
    a(n) = {round(((n+2)/2)^4) - 3*(n+1) + 2}

Formula

a(n) = round(((n+2)/2)^4) - 3*(n+1) + 2.

A331391 Number of binary matrices with a total of n ones, distinct columns each with the same number of ones and nonzero rows in nonincreasing lexicographic order.

Original entry on oeis.org

1, 2, 2, 4, 2, 14, 2, 76, 31, 801, 2, 12797, 2, 233247, 28480, 5560377, 2, 160866915, 2, 5351339038, 193927186, 208746406130, 2, 9342273087807, 5289613, 470405726166256, 4946464287635, 26636935297440055, 2, 1679266767908385729, 2, 116818412262277969513
Offset: 1

Views

Author

Andrew Howroyd, Jan 15 2020

Keywords

Comments

The condition that the rows be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of rows.

Examples

			The a(4) = 4 matrices are:
   [1 0 0 0]   [1]   [1 0]   [1 1]
   [0 1 0 0]   [1]   [1 0]   [1 0]
   [0 0 1 0]   [1]   [0 1]   [0 1]
   [0 0 0 1]   [1]   [0 1]

Crossrefs

Cf. A331126.

Formula

a(n) = Sum{d|n} A331126(n/d, d).
a(p) = 2 for prime p.
Showing 1-9 of 9 results.

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