A306642 -id:A306642 - cp's OEIS frontend

A306641 A(n,k) = Sum_{j=0..n} (kn)!/(j! (n-j)!)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

1, 1, 2, 1, 2, 3, 1, 4, 4, 4, 1, 12, 36, 8, 5, 1, 48, 900, 400, 16, 6, 1, 240, 45360, 94080, 4900, 32, 7, 1, 1440, 3855600, 60614400, 11988900, 63504, 64, 8, 1, 10080, 493970400, 82065984000, 114144030000, 1704214512, 853776, 128, 9

Offset: 0

Seiichi Manyama, Mar 02 2019

Columns are the number of maximal chains to multiples of J in the graded poset of (2 X n) antimagic squares. See Stanley. - Arnav Krishna, Jan 13 2023

			Square array begins:
   1,  1,     1,          1,               1, ...
   2,  2,     4,         12,              48, ...
   3,  4,    36,        900,           45360, ...
   4,  8,   400,      94080,        60614400, ...
   5, 16,  4900,   11988900,    114144030000, ...
   6, 32, 63504, 1704214512, 249344297250048, ...

R. P. Stanley, Enumerative Combinatorics, Vol I, Exercise 53, p. 540.

Seiichi Manyama, Antidiagonals n = 0..49, flattened

Columns 0-3 give A000027(n+1), A000079, A002894, A306642, A345646.
Rows 0-1 give A000012, A208529(n+2).
Main diagonal gives A306644.

A345646 a(n) = Sum_{k=0..n} (4n)! / (k! (n-k)!)^4.

Original entry on oeis.org

1, 48, 45360, 60614400, 114144030000, 249344297250048, 609148118181867264, 1604207350254328934400, 4471935609925802450718000, 13022708340511827298941600000, 39267738740263529465273799855360, 121811974529188978353365962361671680, 386880842128109815466159332537704902400

Offset: 0

Vaclav Kotesovec, Jun 21 2021

nonn

In general, for fixed m >= 1, Sum_{k=0..n} (m*n)! / (k!*(n-k)!)^m ~ (2*m)^(m*n) / (Pi*n)^(m-1).

Seiichi Manyama, Table of n, a(n) for n = 0..279

Column 4 of A306641.

Cf. A306642, A306644.

Table[Sum[(4*n)! / (k! * (n-k)!)^4, {k, 0, n}], {n, 0, 15}]

a(n) ~ 2^(12*n) / (Pi*n)^3.

cp's OEIS Frontend

A306641 A(n,k) = Sum_{j=0..n} (kn)!/(j! (n-j)!)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

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A345646 a(n) = Sum_{k=0..n} (4n)! / (k! (n-k)!)^4.

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cp's OEIS Frontend

A306641 A(n,k) = Sum_{j=0..n} (k*n)!/(j! * (n-j)!)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A345646 a(n) = Sum_{k=0..n} (4*n)! / (k! * (n-k)!)^4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A306641 A(n,k) = Sum_{j=0..n} (kn)!/(j! (n-j)!)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

A345646 a(n) = Sum_{k=0..n} (4n)! / (k! (n-k)!)^4.