A331315 -id:A331315 - cp's OEIS frontend

A331277 Array read by antidiagonals: A(n,k) is the number of binary matrices with k distinct columns and any number of nonzero rows with n ones in every column and columns in decreasing lexicographic order.

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 6, 1, 1, 0, 1, 62, 31, 1, 1, 0, 1, 900, 2649, 160, 1, 1, 0, 1, 16824, 441061, 116360, 841, 1, 1, 0, 1, 384668, 121105865, 231173330, 5364701, 4494, 1, 1, 0, 1, 10398480, 49615422851, 974787170226, 131147294251, 256452714, 24319, 1, 1

Offset: 0

Andrew Howroyd, Jan 13 2020

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

A(n,k) is the number of labeled n-uniform hypergraphs with k edges and no isolated vertices. When n=2 these objects are graphs.

			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    6        62            900        16824      384668 ...
  3 | 1 1   31      2649         441061    121105865 49615422851 ...
  4 | 1 1  160    116360      231173330 974787170226 ...
  5 | 1 1  841   5364701   131147294251 ...
  6 | 1 1 4494 256452714 78649359753286 ...
  ...
The A(2,2) = 6 matrices are:
   [1 0]  [1 0]  [1 0]  [1 1]  [1 0]  [1 0]
   [1 0]  [0 1]  [0 1]  [1 0]  [1 1]  [0 1]
   [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [1 1]
   [0 1]  [0 1]  [1 0]

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=1..3 are A000012, A121251, A136245.
Columns k=0..3 are A000012, A000012, A047665, A137219.
The version with nonnegative integer entries is A331278.
The version with not necessarily distinct columns is A330942.
Cf. A262809 (unrestricted version), A331315, A331639.

T(n,k)={my(m=n*k); sum(j=0, m, binomial(binomial(j,n), k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}

A(n, k) = Sum_{j=0..n*k} binomial(binomial(j,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A262809(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A330942(n, j).
A331639(n) = Sum_{d|n} A(n/d, d).

A331397 Number of nonnegative integer matrices with 2 distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

Original entry on oeis.org

1, 2, 14, 128, 1288, 13472, 143840, 1556480, 17006720, 187208192, 2072948224, 23063920640, 257634273280, 2887544053760, 32456082448384, 365710391902208, 4129672996618240, 46721752249794560, 529486122704568320, 6009576477811539968, 68299997524116635648

Offset: 0

Andrew Howroyd, Jan 15 2020

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=2 of A331315.

Cf. A011782, A052141, A226994, A331396.

seq(n)={Vec(1/(4*sqrt(1 - 12*x + 4*x^2 + O(x*x^n))) + (3 - 4*x)/(4*(1-2*x)))} \\ Andrew Howroyd, Jan 15 2020

a(n) = (A052141(n) + A011782(n))/2.
G.f.: 1/(4*sqrt(1 - 12*x + 4*x^2)) + (3 - 4*x)/(4*(1-2*x)).
a(n) = A011782(n) * A226994(n).
D-finite with recurrence n*a(n) +2*(-8*n+5)*a(n-1) +28*(2*n-3)*a(n-2) +8*(-8*n+19)*a(n-3) +16*(n-3)*a(n-4)=0. - R. J. Mathar, Mar 13 2023

A121227 Number of labeled multigraphs with loops and with n edges and no vertex of degree 0.

Original entry on oeis.org

1, 2, 14, 150, 2210, 41642, 956878, 25955630, 811819826, 28764498386, 1138755852990, 49817190098694, 2386544217733906, 124257113538066522, 6986465328614267742, 421887743219324342110, 27231714819135144778722, 1871047822756547798671074

Offset: 0

Vladeta Jovovic, Sep 06 2006

Seiichi Manyama, Table of n, a(n) for n = 0..364 (terms 0..75 from Nathaniel Johnston)

Row n=2 of A331315.

Unlabeled analog is A007717.

seq(sum(binomial(m*(m+1)/2+n-1,n)/2^(m+1),m=0..infinity),n=0..10);

a(n)={sum(k=0, 2*n, binomial(k*(k+1)/2+n-1, n)*sum(r=k, 2*n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 15 2018

a(n) = Sum_{k=0..2*n} binomial(k*(k+1)/2+n-1, n)*(Sum_{r=k..2*n} binomial(r, k)*(-1)^(r-k)). - Andrew Howroyd, Sep 15 2018

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A331277 Array read by antidiagonals: A(n,k) is the number of binary matrices with k distinct columns and any number of nonzero rows with n ones in every column and columns in decreasing lexicographic order.

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A331397 Number of nonnegative integer matrices with 2 distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

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A121227 Number of labeled multigraphs with loops and with n edges and no vertex of degree 0.

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