A137220 -id:A137220 - cp's OEIS frontend

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.

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

Andrew Howroyd, Jan 13 2020

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.

			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]

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

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.

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 *)

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)))}

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).

A137219 a(n) = (A126086(n) - 3*A001850(n) + 2)/6.

Original entry on oeis.org

1, 62, 2649, 116360, 5364701, 256452714, 12582472897, 629389744448, 31955247002601, 1641724953315062, 85159811841234857, 4452782349569991736, 234393562418967430389, 12409423916979629786322, 660253088667210584565249

Offset: 1

Vladeta Jovovic, Mar 06 2008, Mar 16 2008

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A001850, A047665, A126086, A137220.

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: A137219 := proc(n) (A126086(n)-3*A001850(n)+2)/6 ; end: seq(A137219(n),n=1..30) ; # R. J. Mathar, Apr 01 2008

A126086[n_]:= A126086[n]= Sum[(-1)^k*Binomial[n+k,n]*HypergeometricPFQ[{-k, n+1, n+1}, {1,1}, 1], {k,0,2*n}];
A001850[n_]:= Hypergeometric2F1[-n, n+1, 1, -1];
A137219[n_]:= (A126086[n] - 3*A001850[n] + 2)/6;
Table[A137219[n], {n, 30}] (* G. C. Greubel, Jan 05 2022 *)

def A137219(n): return round( sum( binomial(binomial(j, n), 3)/2^(j+1) for j in (0..1000) ) )
[A137219(n) for n in (1..30)] # G. C. Greubel, Jan 05 2022

a(n) = Sum_{m >= 0} binomial(binomial(m, n), 3)/2^(m+1).

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

cp's OEIS Frontend

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.

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