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 1-4 of 4 results.

A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 3, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 6, 11, 19, 10, 4, 1, 1, 1, 7, 16, 41, 32, 16, 4, 1, 1, 1, 8, 23, 81, 101, 68, 20, 5, 1, 1, 1, 9, 31, 153, 299, 301, 114, 29, 5, 1, 1, 1, 10, 41, 273, 849, 1358, 757, 210, 35, 6, 1
Offset: 0

Views

Author

Andrew Howroyd, May 28 2023

Keywords

Comments

T(n,k) is also the number of equivalence classes of n X k binary matrices with an even number of 1's in each column under permutation of rows and columns.

Examples

			Array begins:
======================================================
n/k| 0 1  2   3    4     5      6       7        8 ...
---+--------------------------------------------------
0  | 1 1  1   1    1     1      1       1        1 ...
1  | 1 1  1   1    1     1      1       1        1 ...
2  | 1 2  3   4    5     6      7       8        9 ...
3  | 1 2  4   7   11    16     23      31       41 ...
4  | 1 3  8  19   41    81    153     273      468 ...
5  | 1 3 10  32  101   299    849    2290     5901 ...
6  | 1 4 16  68  301  1358   6128   27008   114763 ...
7  | 1 4 20 114  757  5567  43534  343656  2645494 ...
8  | 1 5 29 210 1981 23350 319119 4633380 67013431 ...
  ...

Links

Crossrefs

A259344 is the same array without the first row and column read by upward antidiagonals.
Columns k=0..6 are A000012, A004526(n+2), A005232, A006381, A006382, A056204, A056205.
Rows n=2..4 are A000027(n+1), A000601, A006380.
Main diagonal is A006383.
Cf. A028657, A241956, A362905.

Programs

  • PARI
    \\ Compare A028657.
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)={sum(j=1, #q, gcd(t, q[j]))}
T(n, k)={if(n==0, 1, my(s=0); forpart(q=n, my(e=1<

A362906 Number of n element multisets of length 3 vectors over GF(2) that sum to zero.

Original entry on oeis.org

1, 1, 8, 15, 50, 99, 232, 429, 835, 1430, 2480, 3978, 6372, 9690, 14640, 21318, 30789, 43263, 60280, 82225, 111254, 148005, 195416, 254475, 329095, 420732, 534496, 672452, 841160, 1043460, 1287648, 1577532, 1923465, 2330445, 2811240, 3372291, 4029178
Offset: 0

Views

Author

Andrew Howroyd, May 27 2023

Keywords

Comments

a(n) is the number of n X 3 binary matrices under row permutations and column complementations.
See A362905 for other interpretations.

Examples

			The a(1) = 1 multiset is {000}.
The a(2) = 8 multisets are {000, 000}, {001, 001}, {010, 010}, {011, 011}, {100, 100}, {101, 101}, {110, 110}, {111, 111}.
The a(3) = 15 multisets are {000, 000, 000}, {000, 001, 001}, {000, 010, 010}, {000, 011, 011}, {000, 100, 100}, {000, 101, 101}, {000, 110, 110}, {000, 111, 111}, {001, 010, 011}, {001, 100, 101}, {001, 110, 111}, {010, 100, 110}, {010, 101, 111}, {011, 100, 111}, {011, 101, 110}.

Links

Crossrefs

Column k=3 of A362905.
Cf. A006381.

Programs

  • Mathematica
    A362906[n_]:=(Binomial[n+7,7]+If[EvenQ[n],7Binomial[n/2+3,3],0])/8;Array[A362906,50,0] (* Paolo Xausa, Nov 18 2023 *)
  • PARI
    a(n) = (binomial(n+7,7) + if(n%2==0, 7*binomial(n/2+3, 3)))/8

Formula

G.f.: (1 - 3*x + 6*x^2 - 3*x^3 + x^4)/((1 - x)^8*(1 + x)^4).
a(n) = binomial(n+7, 7)/8 for odd n;
a(n) = (binomial(n+7, 7) + 7*binomial(n/2+3, 3))/8 for even n.

A363350 Number of n element multisets of length 4 vectors over GF(2) that sum to zero.

Original entry on oeis.org

1, 1, 16, 51, 276, 969, 3504, 10659, 30954, 81719, 205040, 482885, 1088100, 2340135, 4850640, 9694845, 18789795, 35357670, 64833120, 115997970, 203014680, 347993910, 585292320, 966955410, 1571349780, 2514084066, 3964589856, 6167026726, 9470900056, 14369476066, 21554373984
Offset: 0

Views

Author

Andrew Howroyd, May 30 2023

Keywords

Comments

a(n) is the number of n X 4 binary matrices under row permutations and column complementations.
See A362905 for other interpretations.

Links

Crossrefs

Column k=4 of A362905.
Cf. A006382.

Programs

  • Mathematica
    A363350[n_]:=(Binomial[n+15,15]+If[EvenQ[n],15Binomial[n/2+7,7],0])/16;Array[A363350,50,0] (* Paolo Xausa, Nov 18 2023 *)
  • PARI
    a(n) = (binomial(n+15,15) + if(n%2==0, 15*binomial(n/2+7, 7)))/16

Formula

G.f.: (1 - 7*x + 28*x^2 - 49*x^3 + 70*x^4 - 49*x^5 + 28*x^6 - 7*x^7 + x^8)/((1 - x)^16*(1 + x)^8).
a(n) = binomial(n+15, 15)/16 for odd n;
a(n) = (binomial(n+15, 15) + 15*binomial(n/2+7, 7))/16 for even n.

A363351 Number of n element multisets of length n vectors over GF(2) that sum to zero.

Original entry on oeis.org

1, 1, 4, 15, 276, 11781, 1878976, 1025425687, 1991615557152, 13956142211859705, 356420795746828010496, 33403125520521519582574755, 11550847036800645994553295682560, 14809214844165378046279886451931058885, 70706990798105074752791720424861516970573824
Offset: 0

Views

Author

Andrew Howroyd, May 30 2023

Keywords

Comments

Number of equivalence classes of n X n binary matrices with an even number of 1's in each column under permutation of rows.
Number of equivalence classes of n X n binary matrices under permutation of rows and complementation of columns.

Links

Crossrefs

Main diagonal of A362905.
Cf. A006383.

Programs

  • Mathematica
    A363351[n_]:=(Binomial[2^n+n-1,n]+If[EvenQ[n],(2^n-1)Binomial[2^(n-1)+n/2-1,n/2],0])/2^n;Array[A363351,20,0] (* Paolo Xausa, Nov 19 2023 *)
  • PARI
    a(n)={(binomial(2^n+n-1, n) + if(n%2==0, (2^n-1)*binomial(2^(n-1)+n/2-1, n/2)))/2^n}

Formula

a(n) = binomial(2^n+n-1, n)/2^n for odd n;
a(n) = (binomial(2^n+n-1, n) + (2^n-1)*binomial(2^(n-1)+n/2-1, n/2))/2^n for even n.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.