A137219 -id:A137219 - 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).

A137220 a(n) = (A126086(n) + 3*A001850(n) + 2)/6.

Original entry on oeis.org

1, 4, 75, 2712, 116681, 5366384, 256461703, 12582521536, 629390010177, 31955248465164, 1641724961412515, 85159811886281576, 4452782349821587705, 234393562420377364008, 12409423916987553634575, 660253088667255226947072

Offset: 0

Vladeta Jovovic, Mar 06 2008, Mar 16 2008

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

Column k=3 of A330942.

Cf. A047665, A137219.

A126086 := proc(n) local x,y,z ; coeftayl(coeftayl(coeftayl(1/(1-x-y-z-x*y-x*z-y*z-x*y*z),z=0,n),y=0,n),x=0,n) ; end: A001850 := proc(n) local k ; add(binomial(n,k)*binomial(n+k,k),k=0..n) ; end: A137220 := proc(n) (A126086(n)+3*A001850(n)+2)/6 ; end: seq(A137220(n),n=0..30) ; # R. J. Mathar, Apr 01 2008

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, 3], {n, 0, 15}] (* Jean-François Alcover, Apr 10 2020, after Andrew Howroyd *)

a(n) = {sum(j=0, 3*n, binomial(binomial(j,n)+2, 3) * sum(i=j, 3*n, (-1)^(i-j)*binomial(i,j)))} \\ Andrew Howroyd, Feb 09 2020

@CachedFunction
def A137220(n): return round( -sum( binomial(-binomial(j, n), 3)/2^(j+1) for j in (0..500) ) )
[A137220(n) for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = -Sum_{m>=0} binomial(-binomial(m,n),3)/2^(m+1).
a(n) = A137219(n) + A001850(n). - R. J. Mathar, Apr 01 2008
a(n) = Sum_{j=0..3*n} binomial(binomial(j,n)+2, 3) * (Sum_{i=j..3*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020

More terms from R. J. Mathar, Apr 01 2008

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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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A137220 a(n) = (A126086(n) + 3*A001850(n) + 2)/6.

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