A306641 -id:A306641 - cp's OEIS frontend

A306642 a(n) = Sum_{k=0..n} (3n)!/(k! (n-k)!)^3.

1, 12, 900, 94080, 11988900, 1704214512, 260453217024, 41886697881600, 6996546610936740, 1203384096358158000, 211855235800656848400, 38011289046678107596800, 6928290032159649797280000, 1279703438754969901486464000, 239070018975087493229806080000

Offset: 0

Seiichi Manyama, Mar 02 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..431
Robert W. Donley Jr, Directed path enumeration for semi-magic squares of size three, arXiv:2107.09463 [math.CO], 2021.

Column 3 of A306641.

Array[Sum[(3 #)!/(k!*(# - k)!)^3, {k, 0, #}] &, 15, 0] (* Michael De Vlieger, Dec 02 2021 *)

{a(n) = sum(k=0, n, (3*n)!/(k!*(n-k)!)^3)}

a(n) ~ 216^n / (Pi*n)^2. - Vaclav Kotesovec, Jun 21 2021

A306644 a(n) = Sum_{k=0..n} (n^2)!/(k! * (n-k)!)^n.

Original entry on oeis.org

1, 2, 36, 94080, 114144030000, 128569399991042250240, 231970526672859167062880173363200, 974076884952864555606703666490413198470021120000, 13999785014750877128592398884910508842895938385473568105272652000000

Offset: 0

Seiichi Manyama, Mar 02 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..24

Main diagonal of A306641.

a[n_] := Sum[(n^2)!/(k! * (n-k)!)^n, {k, 0, n}]; Array[a, 9, 0] (* Amiram Eldar, Jun 21 2021 *)

{a(n) = sum(k=0, n, (n^2)!/(k!*(n-k)!)^n)}

a(n) ~ c * 2^(n^2 + 1/2) * n^(n^2 - n + 1) / Pi^(n - 1/2), where c = exp(-1/3)*JacobiTheta3(0, exp(-2)) = exp(-1/3) * EllipticTheta[3, 0, exp(-2)] = 0.910956007080971245990320395256172663671471380838524358269586617628532... if n is even and c = exp(-1/3) * JacobiTheta2(0, exp(-2)) = exp(-1/3) * EllipticTheta[3, 0, exp(-2)] = 0.885121645271745566745223804647879414416684832686710775956467801722557... if n is odd. - Vaclav Kotesovec, Jun 21 2021

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

A306642 a(n) = Sum_{k=0..n} (3n)!/(k! (n-k)!)^3.

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A306644 a(n) = Sum_{k=0..n} (n^2)!/(k! * (n-k)!)^n.

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

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cp's OEIS Frontend

A306642 a(n) = Sum_{k=0..n} (3*n)!/(k! * (n-k)!)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A306644 a(n) = Sum_{k=0..n} (n^2)!/(k! * (n-k)!)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A306642 a(n) = Sum_{k=0..n} (3n)!/(k! (n-k)!)^3.

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