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 1-3 of 3 results.

A094223 Number of binary n X n matrices with all rows (columns) distinct, up to permutation of columns (rows).

Original entry on oeis.org

1, 2, 7, 68, 2251, 247016, 89254228, 108168781424, 451141297789858, 6625037125817801312, 348562672319990399962384, 66545827618461283102105245248, 46543235997095840080425299916917968, 120155975713532210671953821005746669185792, 1152009540439950050422144845158703009569109376384
Offset: 0

Views

Author

Goran Kilibarda and Vladeta Jovovic, May 28 2004

Keywords

Links

Crossrefs

Main diagonal of A059584 and A059587, A060690, A088309.
Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n - k)*StirlingS1[n, k]*Binomial[2^k, n], {k, 0, n}]; (* or *) a[n_] := Sum[ StirlingS1[n, k]*Binomial[2^k + n - 1, n], {k, 0, n}]; Table[ a[n], {n, 0, 12}] (* Robert G. Wilson v, May 29 2004 *)
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k+n-1, n)); \\ Michel Marcus, Dec 17 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*binomial(2^k, n).
a(n) = Sum_{k=0..n} Stirling1(n, k)*binomial(2^k+n-1, n).

Extensions

More terms from Robert G. Wilson v, May 29 2004
a(13) onwards from Andrew Howroyd, Jan 20 2024

A059585 Number of labeled 3-node T_0-hypergraphs with n hyperedges (empty hyperedges and multiple hyperedges included).

Original entry on oeis.org

0, 0, 12, 68, 235, 636, 1478, 3088, 5958, 10800, 18612, 30756, 49049, 75868, 114270, 168128, 242284, 342720, 476748, 653220, 882759, 1178012, 1553926, 2028048, 2620850, 3356080, 4261140, 5367492, 6711093, 8332860, 10279166, 12602368
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Jan 23 2001

Keywords

Comments

A hypergraph is a T_0 hypergraph if for every two distinct nodes there exists a hyperedge containing one but not the other node.

Links

Crossrefs

Cf. A059084, a(n)=A059584(n, 3), A059586.

Programs

  • Magma
    [n*(n-1)*(n+1)*(n^4+28*n^3+323*n^2+1988*n+ 4572)/5040: n in [0..35]]; // Vincenzo Librandi, Oct 07 2017
  • Maple
    for n from 0 to 100 do printf(`%d,`,n*(n - 1)*(n + 1)*(n^4 + 28*n^3 + 323*n^2 + 1988*n + 4572)/5040) od:
  • Mathematica
    CoefficientList[Series[x^2*(2 - x)^2*(3 - 4*x + 2*x^2)/(1 - x)^8, {x, 0, 50}], x] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 12, 68, 235, 636, 1478, 3088}, 33] (* Vincenzo Librandi, Oct 07 2017 *)
  • PARI
    x='x+O('x^50); concat([0,0],Vec(x^2*(2-x)^2*(3-4*x+2*x^2)/(1-x)^8)) \\ G. C. Greubel, Oct 06 2017

Formula

a(n) = binomial(n + 7, n) - 3*binomial(n + 3, n) + 2*binomial(n + 1, n) = n*(n - 1)*(n + 1)*(n^4 + 28*n^3 + 323*n^2 + 1988*n + 4572)/5040.
G.f.: x^2*(2-x)^2*(3-4*x+2*x^2)/(1-x)^8. - Colin Barker, Jun 25 2012

Extensions

More terms from James Sellers, Jan 24 2001

A059586 Number of labeled T_0-hypergraphs with n hyperedges (empty hyperedges and multiple hyperedges included).

Original entry on oeis.org

2, 5, 35, 18301, 2369751675482, 5960531437867327674550533616796025, 479047836152505670895481842190009123676957243077039723706127824160370689849840668444493
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Jan 23 2001

Keywords

Comments

A hypergraph is a T_0 hypergraph if for every two distinct nodes there exists a hyperedge containing one but not the other node.

Examples

			a(3) = (1/3!) * (2 * [2! * e] + 3 * [4! * e] + [8! * e]) = (1/3!) * (2 * 5 + 3 * 65 + 109601) = 18301, where [k! * e] := floor(k! * exp(1)).

Crossrefs

Cf. A000522, A059086, A059584, A059585.

Programs

  • Maple
    with(combinat): Digits := 1000: f := n->(1/n!)*sum(abs(stirling1(n,i))*floor((2^i)!*exp(1)), i=0..n): for n from 0 to 8 do printf(`%d,`,f(n)) od:

Formula

a(n)=(1/n!)*Sum_{k=0..n} |stirling1(n, k)|*floor((2^k)!*exp(1)).

Extensions

More terms from James Sellers, Jan 24 2001
Showing 1-3 of 3 results.

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