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 1-10 of 11 results. Next

A107864 Duplicate of A099121.

Original entry on oeis.org

1, 3, 21, 220, 3060, 53130, 1107568, 26978328, 752538150, 23667689815, 828931106355, 32006008361808, 1350990969850340, 61902409203193230, 3060335715568296000, 162392216278033616560, 9206887338937200407418
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

A107862 Triangle, read by rows, where T(n,k) = C(n*(n-1)/2 - k*(k-1)/2 + n-k, n-k).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 20, 10, 3, 1, 210, 84, 21, 4, 1, 3003, 1001, 220, 36, 5, 1, 54264, 15504, 3060, 455, 55, 6, 1, 1184040, 296010, 53130, 7315, 816, 78, 7, 1, 30260340, 6724520, 1107568, 142506, 14950, 1330, 105, 8, 1, 886163135, 177232627, 26978328, 3262623, 324632, 27405, 2024, 136, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

Comments

Remarkably, the following matrix products are all equal to A107876: A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873.

Examples

			Triangle begins:
        1;
        1,      1;
        3,      2,     1;
       20,     10,     3,    1;
      210,     84,    21,    4,   1;
     3003,   1001,   220,   36,   5,  1;
    54264,  15504,  3060,  455,  55,  6, 1;
  1184040, 296010, 53130, 7315, 816, 78, 7, 1; ...

Links

Crossrefs

Cf. A014068 (column 0), A107863 (column 1), A099121 (column 2), A107865, A107867, A107870, A107876.

Programs

  • Magma
    [Binomial(Floor((n-k)*(n+k+1)/2), n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2022
  • Mathematica
    T[n_,k_]:= Binomial[(n-k)*(n+k+1)/2, n-k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 19 2022 *)
  • PARI
    T(n,k)=binomial(n*(n-1)/2-k*(k-1)/2+n-k,n-k)
  • Sage
    flatten([[binomial( (n-k)*(n+k+1)/2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2022

Formula

T(n,k) = binomial( (n-k)*(n+k+1)/2, n-k). - G. C. Greubel, Feb 19 2022

A107867 Triangle, read by rows, where T(n,k) = C(n*(n-1)/2-k*(k-1)/2+n-k+1,n-k).

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 35, 15, 4, 1, 330, 120, 28, 5, 1, 4368, 1365, 286, 45, 6, 1, 74613, 20349, 3876, 560, 66, 7, 1, 1560780, 376740, 65780, 8855, 969, 91, 8, 1, 38608020, 8347680, 1344904, 169911, 17550, 1540, 120, 9, 1, 1101716330, 215553195, 32224114
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

Comments

Remarkably, the following matrix products are all equal to A107876: A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873.

Examples

			Triangle begins:
1;
2,1;
6,3,1;
35,15,4,1;
330,120,28,5,1;
4368,1365,286,45,6,1;
74613,20349,3876,560,66,7,1;
1560780,376740,65780,8855,969,91,8,1; ...

Crossrefs

Cf. A099121, A107862, A107865, A107867, A107870, A107876.

Programs

  • PARI
    T(n,k)=binomial(n*(n-1)/2-k*(k-1)/2 +n-k+1,n-k)

A331436 Array read by antidiagonals: A(n,k) is the number of n element multisets of n element multisets of a k-set.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 21, 20, 1, 0, 1, 5, 55, 220, 70, 1, 0, 1, 6, 120, 1540, 3060, 252, 1, 0, 1, 7, 231, 7770, 73815, 53130, 924, 1, 0, 1, 8, 406, 30856, 1088430, 5461512, 1107568, 3432, 1, 0, 1, 9, 666, 102340, 11009376, 286243776, 581106988, 26978328, 12870, 1, 0
Offset: 0

Views

Author

Andrew Howroyd, Jan 17 2020

Keywords

Examples

			Array begins:
==================================================================
n\k | 0 1   2       3         4            5              6
----+-------------------------------------------------------------
  0 | 1 1   1       1         1            1              1 ...
  1 | 0 1   2       3         4            5              6 ...
  2 | 0 1   6      21        55          120            231 ...
  3 | 0 1  20     220      1540         7770          30856 ...
  4 | 0 1  70    3060     73815      1088430       11009376 ...
  5 | 0 1 252   53130   5461512    286243776     8809549056 ...
  6 | 0 1 924 1107568 581106988 127860662755 13949678575756 ...
    ...
The A(2,2) = 6 multisets are:
   {{1,1}, {1,1}},
   {{1,1}, {1,2}},
   {{1,1}, {2,2}},
   {{1,2}, {1,2}},
   {{1,2}, {2,2}},
   {{2,2}, {2,2}}.

Links

Crossrefs

Rows n=0..3 are A000012, A001477, A002817, A140236.
Columns k=0..10 are A000007, A000012, A000984, A099121, A099122, A099123, A099124, A099125, A099126, A099127, A099128.
Min diagonal is A331477.

Programs

  • PARI
    T(n,k)={binomial(binomial(n + k - 1, n) + n - 1, n)}
{ for(n=0, 7, for(k=0, 7, print1(T(n,k), ", ")); print) }

Formula

A(n,k) = binomial(binomial(n + k - 1, n) + n - 1, n).

A099122 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3}.

Original entry on oeis.org

1, 4, 55, 1540, 73815, 5461512, 581106988, 84431259000, 16104878212995, 3910294246315600, 1178924607035010836, 432472873725488656424, 189789513537655207705620, 98222259182333060014344720
Offset: 0

Views

Author

Sascha Kurz, Sep 28 2004

Keywords

Comments

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 3, where the referees cannot be distinguished.
a(n) is the number n element multisets of n element multisets of a 4-set. - Andrew Howroyd, Jan 17 2020

Links

Crossrefs

Column k=4 of A331436.
Cf. A099121, A099123, A099124, A099125, A099126, A099127, A099128.

Programs

  • PARI
    a(n)={binomial(binomial(n+3, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

Formula

a(n) = binomial(binomial(n+3, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

Extensions

a(0)=1 prepended by Andrew Howroyd, Jan 17 2020

A099123 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4}.

Original entry on oeis.org

1, 5, 120, 7770, 1088430, 286243776, 127860662755, 90079147136880, 94572327271677750, 141504997346476482290, 291098519807782284023426, 799388312264077003441393875, 2859142263297618955891805452700
Offset: 0

Views

Author

Sascha Kurz, Sep 28 2004

Keywords

Comments

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 4, where the referees cannot be distinguished.
a(n) is the number of n element multisets of n element multisets of a 5-set. - Andrew Howroyd, Jan 17 2020

Links

Crossrefs

Column k=5 of A331436.
Cf. A099121, A099122, A099124, A099125, A099126, A099127, A099128.

Programs

  • PARI
    a(n)={binomial(binomial(n + 4, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

Formula

a(n) = binomial(binomial(n + 4, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

Extensions

a(0)=1 prepended by Andrew Howroyd, Jan 17 2020

A099124 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5}.

Original entry on oeis.org

1, 6, 231, 30856, 11009376, 8809549056, 13949678575756, 39822612151165272, 190782296093487153627, 1449479533445348118223510, 16683660613067331275158983216, 280167196060745030529247396914000, 6651137552302201488023930244802896266
Offset: 0

Views

Author

Sascha Kurz, Sep 28 2004

Keywords

Comments

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 5, where the referees cannot be distinguished.
a(n) is the number of n element multisets of n element multisets of a 6-set. - Andrew Howroyd, Jan 17 2020

Links

Crossrefs

Column k=6 of A331436.
Cf. A099121, A099122, A099123, A099125, A099126, A099127, A099128.

Programs

  • Mathematica
    Table[Binomial[Binomial[n+5,n]+n-1,n],{n,0,20}] (* Harvey P. Dale, Jul 26 2020 *)
  • PARI
    a(n)={binomial(binomial(n + 5, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

Formula

a(n) = binomial(binomial(n + 5, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

Extensions

a(0)=1 prepended and a(12) and beyond from Andrew Howroyd, Jan 17 2020

A099125 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6}.

Original entry on oeis.org

1, 7, 406, 102340, 83369265, 179224992408, 878487565272240, 8800321588119330984, 165564847349896309234920, 5470105884755875924791320090, 300550263698274781577833262263448, 26251679033395309424785182716562495776, 3509663406416043297299781592276029113718775
Offset: 0

Views

Author

Sascha Kurz, Sep 28 2004

Keywords

Comments

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 6, where the referees cannot be distinguished.
a(n) is the number of n element multisets of n element multisets of a 7-set. - Andrew Howroyd, Jan 17 2020

Links

Crossrefs

Column k=7 of A331436.
Cf. A099121, A099122, A099123, A099124, A099126, A099127, A099128.

Programs

  • PARI
    a(n)={binomial(binomial(n + 6, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

Formula

a(n) = binomial(binomial(n + 6, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

Extensions

a(0)=1 prepended and a(11) and beyond from Andrew Howroyd, Jan 17 2020

A099126 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6,7}.

Original entry on oeis.org

1, 8, 666, 295240, 503167995, 2629770332904, 35773664992355004, 1119582594247762626696, 73241437035618231162682185, 9277639855710782695858431981840, 2137918570337064383107929197622033920, 850936582591338109213109187016928388683280
Offset: 0

Views

Author

Sascha Kurz, Oct 11 2004

Keywords

Comments

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 7, where the referees cannot be distinguished.
a(n) is the number of n element multisets of n element multisets of an 8-set. - Andrew Howroyd, Jan 17 2020

Links

Crossrefs

Column k=8 of A331436.
Cf. A099121, A099122, A099123, A099124, A099125, A099127, A099128.

Programs

  • PARI
    a(n)={binomial(binomial(n + 7, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

Formula

a(n) = binomial(binomial(n + 7, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

Extensions

a(0)=1 prepended and a(11) and beyond from Andrew Howroyd, Jan 17 2020

A099127 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6,7,8}.

Original entry on oeis.org

1, 9, 1035, 762355, 2531986380, 29653914688398, 1023687680214527328, 90954904732217610881940, 18709083803797153776767847375, 8183604949527627465377060678018870, 7099997495119970047949715137555520213198
Offset: 0

Views

Author

Sascha Kurz, Oct 11 2004

Keywords

Comments

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 8, where the referees cannot be distinguished.
a(n) is the number of n element multisets of n element multisets of a 9-set. - Andrew Howroyd, Jan 17 2020

Links

Crossrefs

Column k=9 of A331436.
Cf. A099121, A099122, A099123, A099124, A099125, A099126, A099128.

Programs

  • PARI
    a(n)={binomial(binomial(n + 8, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

Formula

a(n) = binomial(binomial(n + 8, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

Extensions

a(0)=1 prepended and a(10) and beyond from Andrew Howroyd, Jan 17 2020
Showing 1-10 of 11 results. Next

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

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