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.

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A066991 Square array read by descending antidiagonals of number of ways of dividing n*k labeled items into k unlabeled orders with n items in each order.

Original entry on oeis.org

1, 1, 2, 1, 12, 6, 1, 120, 360, 24, 1, 1680, 60480, 20160, 120, 1, 30240, 19958400, 79833600, 1814400, 720, 1, 665280, 10897286400, 871782912000, 217945728000, 239500800, 5040, 1, 17297280, 8892185702400, 20274183401472000
Offset: 1

Views

Author

Henry Bottomley, Feb 01 2002

Keywords

Comments

T(p,k) = (pk)!/k! is divisible by p^k but not p^(k+1) for p prime; e.g., T(3,4) = 3^4*11*10*8*7*5*4*2*1 = 19958400.

Examples

			The array begins:
  n\k|   1        2             3                   4  ...
  --------------------------------------------------------
   1 |   1,       1,            1,                  1, ...
   2 |   2,      12,          120,               1680, ...
   3 |   6,     360,        60480,           19958400, ...
   4 |  24,   20160,     79833600,       871782912000, ...
   5 | 120, 1814400, 217945728000, 101370917007360000, ...
  ...

Links

Crossrefs

Rows include A000012, A001813, A064350.
Columns include A000142, A002674, A065961.
Cf. A060538, A060539, A060540.

Programs

  • Mathematica
    Table[((n-k+1)*k)!/k!, {n, 10}, {k, n, 1, -1}] (* Paolo Xausa, Feb 19 2024 *)

Formula

T(n,k) = (n*k)!/k!.

Extensions

Edited by Paolo Xausa, Feb 19 2024

A137645 a(n) = Sum_{k=0..n} C((n-k)*k, k) * C((n-k)*k, n-k).

Original entry on oeis.org

1, 0, 1, 4, 42, 608, 10986, 240492, 6167112, 181154848, 5995624710, 220711502648, 8943846698096, 395588177834784, 18962600075658460, 979198125493716492, 54189002212286942316, 3199366560075461850320, 200730550064907653703510, 13336507142191259122442532, 935401326531455246646557760, 69066745767857553528070539760, 5355032622687046432711489319940
Offset: 0

Views

Author

Paul D. Hanna, Jan 31 2008

Keywords

Examples

			The initial terms of this sequence are
a(0) = 1 = 1*1;
a(1) = 0 = 1*0 + 0*1;
a(2) = 1 = 1*0 + 1*1 + 0*1;
a(3) = 4 = 1*0 + 2*1 + 1*2 + 0*1;
a(4) = 42 = 1*0 + 3*1 + 6*6 + 1*3 + 0*1;
a(5) = 608 = 1*0 + 4*1 + 15*20 + 20*15 + 1*4 + 0*1;
a(6) = 10986 = 1*0 + 5*1 + 28*70 + 84*84 + 70*28 + 1*5 + 0*1;
a(7) = 240492 = 1*0 + 6*1 + 45*252 + 220*495 + 495*220 + 252*45 + 1*6 + 0*1; ...
where the triangle of coefficients binomial((n-k)*k, k) begins:
1;
1, 0;
1, 1, 0;
1, 2, 1, 0;
1, 3, 6, 1, 0;
1, 4, 15, 20, 1, 0;
1, 5, 28, 84, 70, 1, 0;
1, 6, 45, 220, 495, 252, 1, 0;
1, 7, 66, 455, 1820, 3003, 924, 1, 0;
1, 8, 91, 816, 4845, 15504, 18564, 3432, 1, 0;
1, 9, 120, 1330, 10626, 53130, 134596, 116280, 12870, 1, 0; ...
and the triangle A060539 of coefficients binomial((n-k)*k, n-k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 6, 3, 1;
0, 1, 20, 15, 4, 1;
0, 1, 70, 84, 28, 5, 1;
0, 1, 252, 495, 220, 45, 6, 1;
0, 1, 924, 3003, 1820, 455, 66, 7, 1;
0, 1, 3432, 18564, 15504, 4845, 816, 91, 8, 1;
0, 1, 12870, 116280, 134596, 53130, 10626, 1330, 120, 9, 1; ...

Links

Crossrefs

Cf. A060539.

Programs

  • PARI
    {a(n)=sum(k=0,n,binomial((n-k)*k,k)*binomial((n-k)*k,n-k))}
for(n=0,25,print1(a(n),", "))

A184357 a(n) = Sum_{k=0..n} C(n^2-k^2, n-k)*C(k^2, k).

Original entry on oeis.org

1, 2, 15, 226, 5079, 151326, 5611906, 248995090, 12862665297, 758353907422, 50255751919386, 3698524145800452, 299324750430958973, 26424096787968560864, 2527130527406877225450, 260305991718814269022586, 28732428200125730917353569
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2011

Keywords

Examples

			a(0) = 1 = 1*1;
a(1) = 2 = 1*1 + 1*1;
a(2) = 15 = 6*1 + 3*1 + 1*6;
a(3) = 226 = 84*1 + 28*1 + 5*6 + 1*84;
a(4) = 5079 = 1820*1 + 455*1 + 66*6 + 7*84 + 1*1820;
a(5) = 151326 = 53130*1 + 10626*1 + 1330*6 + 120*84 + 9*1820 + 1*53130; ...

Links

Crossrefs

Cf. A060539, A014062.

Programs

  • Mathematica
    Table[Sum[Binomial[n^2-k^2,n-k]Binomial[k^2,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 30 2023 *)
  • PARI
    {a(n)=if(n<0, 0, sum(k=0, n, binomial(n^2-k^2, n-k)*binomial(k^2, k)))}
Previous Showing 11-13 of 13 results.

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