A221177 -id:A221177 - cp's OEIS frontend

A141906 Triangle t(n,m) = (n*m)!/(m!^n) read by rows, 0<=m<=n.

1, 1, 1, 1, 2, 6, 1, 6, 90, 1680, 1, 24, 2520, 369600, 63063000, 1, 120, 113400, 168168000, 305540235000, 623360743125120, 1, 720, 7484400, 137225088000, 3246670537110000, 88832646059788350720, 2670177736637149247308800

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 14 2008

Row sums are in A221177.

			1;
1, 1;
1, 2, 6;
1, 6, 90, 1680;
1, 24, 2520, 369600, 63063000;
1, 120, 113400, 168168000, 305540235000, 623360743125120;
1, 720, 7484400, 137225088000, 3246670537110000, 88832646059788350720, 2670177736637149247308800;

Seiichi Manyama, Rows n = 0..26, flattened

Cf. A034841, A187783, A221177.

A141906 := proc(n,m)
        (n*m)!/m!^n ;
end proc:
seq(seq(A141906(n,m),m=0..n),n=0..5) ; # R. J. Mathar, Nov 08 2011

Clear[t, n, m]; t[n_, m_] = (n*m)!/m!^n; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

A306241 a(n) = Sum_{k=0..n} (k*n)!/n!^k.

Original entry on oeis.org

1, 2, 8, 1702, 63097722, 623372476627154, 2670179107513625597282318, 7363615751879726008424500256018442794, 18165723935734974232438957032838329596079311234990642

Offset: 0

Seiichi Manyama, Jan 31 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A034841, A120666, A221177.

Table[Sum[(k*n)!/n!^k,{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Feb 08 2019 *)

PARI
```
{a(n) = sum(k=0, n, (k*n)!/n!^k)}
```

a(n) equals (row sums of A120666) + 1.
From Vaclav Kotesovec, Feb 08 2019: (Start)
a(n) ~ A034841(n).
a(n) ~ n^(n^2 - n/2 + 1) / (exp(1/12) * (2*Pi)^((n-1)/2)). (End)

A342107 a(n) = Sum_{k=0..n} (4*k)!/k!^4.

Original entry on oeis.org

1, 25, 2545, 372145, 63435145, 11796180169, 2320539673225, 474838887231625, 100035931337622625, 21552788197602942625, 4726913659271173170145, 1051798742538350304851425, 236861100204680963085573025

Offset: 0

Seiichi Manyama, Feb 28 2021

Partial sums of A008977.

In general, for m > 1, Sum_{k=0..n} (m*k)!/k!^m ~ m^(m*n + m + 1/2) / ((m^m - 1) * (2*Pi*n)^((m-1)/2)). - Vaclav Kotesovec, Feb 28 2021

Cf. A006134, A008977, A188441, A221177.

A342107 := proc(n)
    add((4*k)!/k!^4,k=0..n) ;
end proc:
seq(A342107(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

Table[Sum[(4*k)!/k!^4, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 28 2021 *)

PARI
```
a(n) = sum(k=0, n, (4*k)!/k!^4);
```

a(n) ~ 2^(8*n + 15/2) / (255 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Feb 28 2021

D-finite with recurrence n^3*a(n) +(-257*n^3+384*n^2-176*n+24)*a(n-1) +8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Dec 04 2023

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A141906 Triangle t(n,m) = (n*m)!/(m!^n) read by rows, 0<=m<=n.

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A306241 a(n) = Sum_{k=0..n} (k*n)!/n!^k.

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A342107 a(n) = Sum_{k=0..n} (4*k)!/k!^4.

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