A096126 -id:A096126 - cp's OEIS frontend

A060540 Square array read by antidiagonals downwards: T(n,k) = (nk)!/(k!^nn!), (n>=1, k>=1), the number of ways of dividing nk labeled items into n unlabeled boxes with k items in each box.

1, 1, 1, 1, 3, 1, 1, 10, 15, 1, 1, 35, 280, 105, 1, 1, 126, 5775, 15400, 945, 1, 1, 462, 126126, 2627625, 1401400, 10395, 1, 1, 1716, 2858856, 488864376, 2546168625, 190590400, 135135, 1, 1, 6435, 66512160, 96197645544, 5194672859376, 4509264634875, 36212176000, 2027025, 1

Offset: 1

Henry Bottomley, Apr 02 2001

The Copeland link gives the associations of this entry with the operator calculus of Appell Sheffer polynomials, the combinatorics of simple set partitions encoded in the Faa di Bruno formula for composition of analytic functions (formal Taylor series), the Pascal matrix, and the geometry of the n-dimensional simplices (hypertriangles, or hypertetrahedra). These, in turn, are related to simple instances of the application of the exponential formula / principle / schema giving the number of not-necessarily-connected objects composed from an ensemble of connected objects. - Tom Copeland, Jun 09 2021

			Array begins:
  1,   1,       1,          1,             1,                 1, ...
  1,   3,      10,         35,           126,               462, ...
  1,  15,     280,       5775,        126126,           2858856, ...
  1, 105,   15400,    2627625,     488864376,       96197645544, ...
  1, 945, 1401400, 2546168625, 5194672859376, 11423951396577720, ...
  ...

Seiichi Manyama, Antidiagonals n = 1..50, flattened (first 20 antidiagonals from Harry J. Smith)
Tom Copeland, Calculus, Combinatorics, and Geometry Underlying OEIS A060540, and the Exponential Formula, 2021.
Nattawut Phetmak and Jittat Fakcharoenphol, Uniformly Generating Derangements with Fixed Number of Cycles in Polynomial Time, Thai J. Math. (2023) Vol. 21, No. 4, 899-915. See pp. 901, 914.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 13.

Rows include A000012, A001700, A060542, A082368, A322252.
Columns k=1..10 give A000012, A001147, A025035, A025036, A025037, A025038, A025040, A025041, A025042.
Main diagonal is A057599.
Related to A057599, see also A096126 and A246048.
Cf. A060358, A361948 (includes row/col 0).
Cf. A000217, A000292, A000332, A000389, A000579, A000580, A007318, A036040, A099174, A133314, A132440, A135278 (associations in Copeland link).

T[n_, k_] := (n*k)!/(k!^n*n!);
Table[T[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 29 2018 *)

{ i=0; for (m=1, 20, for (n=1, m, k=m - n + 1; write("b060540.txt", i++, " ", (n*k)!/(k!^n*n!))); ) } \\ Harry J. Smith, Jul 06 2009

T(n,k) = (n*k)!/(k!^n*n!) = T(n-1,k)*A060543(n,k) = A060538(n,k)/k!.

T(n,k) = Product_{j=2..n} binomial(j*k-1,k-1). - M. F. Hasler, Aug 22 2014

Definition reworded by M. F. Hasler, Aug 23 2014

A096127 a(n) is the largest k such that (n^2)!/(n!)^k is an integer.

Original entry on oeis.org

3, 4, 5, 6, 8, 8, 9, 10, 12, 12, 14, 14, 16, 18, 17, 18, 20, 20, 22, 24, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 33, 35, 36, 38, 38, 38, 40, 42, 42, 42, 44, 44, 46, 48, 48, 48, 50, 50, 52, 54, 55, 54, 56, 58, 58, 60, 60, 60, 62, 62, 64, 66, 65, 67, 68, 68, 70, 72, 73, 72, 74, 74

Offset: 2

Amarnath Murthy, Jul 03 2004

nonn

Conjecture: a(n)=n+1 only when n is prime or a power of a prime. [Verified for n=2..5000. - Amiram Eldar, Apr 06 2021]

			a(6) = 8 as 36!/(6!)^8 is an integer which is not further divisible by 720.

Amiram Eldar, Table of n, a(n) for n = 2..5000

Cf. A034841, A057599, A096126.

f[n_] := Block[{k = n}, While[ IntegerQ[(n^2)!/n!^k], k++ ]; k - 1]; Table[ f[n], {n, 75}] (* Robert G. Wilson v, Jul 03 2004 *)

Edited by Don Reble and Robert G. Wilson v, Jul 04 2004

A246048 Numbers for which (n^2)! is divisible by n!^n*(2n)!.

Original entry on oeis.org

6, 12, 14, 15, 21, 22, 24, 26, 28, 30, 35, 38, 39, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 69, 70, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99

Offset: 1

M. F. Hasler, Aug 23 2014

nonn

In general, (n*m)! is divisible by m!^n*n!, cf. A060540 for the quotients. It was asked when it is also divisible by m!^n*(kn)! for some k>1. The present sequence answers this for the special case m=n. For the values m=n=52,69,75,77,78,92,95,... one can take k=3; m=n=120 is the least case where one can take k=4.

Farideh Firoozbakht observes that all terms are composite numbers. The comment in A057599 and conjecture in A096126 seem to confirm that there are no primes nor powers of primes in this sequence.

max_k(n)=for(k=1,m=n,Mod((n*m)!,m!^n*(k*n)!) && return(k-1)) \\ returns the maximal k for m=n.
for(n=1,99,a(n)>1&&print1(n,",")) \\ prints this sequence

A244443 Smallest integer m > 1 such that m!^(m + n) divides (m^2)!.

Original entry on oeis.org

2, 6, 15, 77, 185, 187, 475, 3820, 4043, 4090, 11231, 30589, 57023, 126815, 131055, 983032, 983033, 2617339, 4046839, 11534206, 11534207, 65011702, 66777087, 368279551, 469745405, 973061887, 1064828671

Offset: 1

Farideh Firoozbakht, Aug 24 2014

The constraint m > 1 is necessary because (1^2)! = 1.
The motivation for this sequence came from comments on the sequence A246048 by M. F. Hasler.
The integer (3820^2)!/(3820!)^3828 related to a(8) has 52166326 digits, so it isn't easy to find more terms.
a(28) > 1.5 * 10^9. - Hiroaki Yamanouchi, Sep 29 2014

			a(4) = 77 because 77!^(77 + 4) divides (77^2)! and 77 is the smallest integer m, m > 1, with this property.

Cf. A096126, A096127, A246048.

for(n=1, 7, m=2; while((m^2)!%(m!^(m+n)), m++); print1(m", ")) \\ Jens Kruse Andersen, Aug 31 2014

n=f=1; for(m=2, 5000, f*=m; s=m^2; forprime(p=2, m, e=0; b=p; while(b<=s, e+=s\b; b*=p); if(valuation(f,p)*(m+n)>e, next(2))); print1(m", "); n++) \\ Faster program. Jens Kruse Andersen, Aug 31 2014

a(9)-a(13) from Jens Kruse Andersen, Aug 31 2014

a(14)-a(27) from Hiroaki Yamanouchi, Sep 29 2014

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A060540 Square array read by antidiagonals downwards: T(n,k) = (n*k)!/(k!^n*n!), (n>=1, k>=1), the number of ways of dividing nk labeled items into n unlabeled boxes with k items in each box.

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A096127 a(n) is the largest k such that (n^2)!/(n!)^k is an integer.

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Mathematica

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A246048 Numbers for which (n^2)! is divisible by n!^n*(2n)!.

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A244443 Smallest integer m > 1 such that m!^(m + n) divides (m^2)!.

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Author

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Comments

Examples

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Programs

PARI

PARI

Extensions

A060540 Square array read by antidiagonals downwards: T(n,k) = (nk)!/(k!^nn!), (n>=1, k>=1), the number of ways of dividing nk labeled items into n unlabeled boxes with k items in each box.