A057599 -id:A057599 - 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

A096126 a(n) is the least integer of the form (n^2)!/(n!)^k.

Original entry on oeis.org

1, 3, 280, 2627625, 5194672859376, 5150805819130303332, 1461034854396267778567973305958400, 450538787986875167583433232345723106006796340625, 146413934927214422927834111686633731590253260933067148964500000000

Offset: 1

Amarnath Murthy, Jul 03 2004

nonn

(n^2)!/(n!)^(n+1) is an integer for every n (see A057599). Hence k >= n+1. Conjecture: k=n+1 only when n is prime or a power of a prime.

			a(4) = 16!/(4!)^5 = 2627625 which is not further divisible by 24.

Andrew Howroyd, Table of n, a(n) for n = 1..25

Cf. A034841, A057599, A096127.

a(n)={if(n==1, 1, (n^2)!/(n!^valuation((n^2)!,n!)))} \\ Andrew Howroyd, Nov 09 2019

Edited by Don Reble, Jul 04 2004

a(9) from Andrew Howroyd, Nov 09 2019

A068740 Result after dividing (n^n)! as many times as possible by n!.

Original entry on oeis.org

1, 1, 3, 833712928048000000

Offset: 0

Henry Bottomley, Feb 26 2002

nonn

For prime n, it is also the number of generalized knockout tournament seedings with n players in one match and n rounds (see formula below). - Alexander Karpov, Dec 14 2017
Next term is too large to include.
From Robert G. Wilson v, Dec 14 2017: (Start)
a(4) = 4125147631...      (370 digits)...3291015625,
a(5) = 3483655217...     (7923 digits)...3819109376,
a(6) = 2196422024...   (164237 digits)...0161431552,
a(7) = 4948281440...  (4005981 digits)...0000000000,
a(8) = 4242413765...(102886160 digits)...4619140625,
(End)

			a(3)=833712928048000000 since 3!=6 and (3^3)!=27!=10888869450418352160768000000 which is divisible by 6^13=13060694016 but not 6^14=78364164096.

Robert G. Wilson v, Table of n, a(n) for n = 0..4
Alexander Karpov, Generalized knockout tournaments, National Research University Higher School of Economics. WP7/2017/03.

Cf. A000142, A000312, A023037, A057599, A068741, A068742.

a(n) = A068741(n)/A068742(n).

For p prime, a(p) = (p^p)!/(p!)^((p^p-1)/(p-1)).

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

A370364 Number of partitions of [n^2] into n sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

Original entry on oeis.org

0, 1, 1, 28, 22893, 2443061876, 68542265471953355, 833412961429901104030214430, 6514551431426932053792271970458170132097, 45458343253887079540702419310885199704811913950207054152, 375236832464739513549091449370258959406125572044428827214970469920572831639

Offset: 0

Alois P. Heinz, Feb 16 2024

nonn

			a(1) = 1: 1.
a(2) = 1: 12|34.
a(3) = 28: 123|456|789, 123|457|689, 123|458|679, 123|459|678, 123|467|589, 123|468|579, 123|469|578, 123|478|569, 123|479|568, 123|489|567, 124|356|789, 125|346|789, 126|345|789, 127|389|456, 128|379|456, 129|378|456, 134|256|789, 135|246|789, 136|245|789, 137|289|456, 138|279|456, 139|278|456, 145|236|789, 146|235|789, 156|234|789, 178|239|456, 179|238|456, 189|237|456.

Alois P. Heinz, Table of n, a(n) for n = 0..27
Wikipedia, Partition of a set

Main diagonal of A370363.

Cf. A057599, A370367.

a:= n-> add((-1)^(n-j+1)*binomial(n, j)*(n*j)!/(j!*n!^j), j=0..n-1):
seq(a(n), n=0..10);

a(n) = Sum_{j=0..n-1} (-1)^(n-j+1)*binomial(n,j)*(n*j)!/(j!*n!^j).
a(n) = A370363(n,n).
a(n) = A057599(n) - A370367(n).

A370367 Number of partitions of [n^2] into n sets of size n having no set of consecutive numbers whose maximum (if k>n) is a multiple of n.

Original entry on oeis.org

1, 0, 2, 252, 2604732, 5192229797500, 3708511647508346445685, 1461034020983306348666869275743970, 450538781472323736156501178553451135548626208528, 146413934881756079673947032145931312279368061228255235014292945848

Offset: 0

Alois P. Heinz, Feb 16 2024

nonn

			a(2) = 2: 13|24, 14|23.

Alois P. Heinz, Table of n, a(n) for n = 0..27
Wikipedia, Partition of a set

Main diagonal of A370366.

Cf. A057599, A370364.

a:= n-> add((-1)^(n-j)*binomial(n, j)*(n*j)!/(j!*n!^j), j=0..n):
seq(a(n), n=0..10);

a(n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(n*j)!/(j!*n!^j).
a(n) = A370366(n,n).
a(n) = A057599(n) - A370364(n).

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

A361948 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} binomial((j + 1)*n - 1, n - 1) if n >= 1, and A(0, k) = 1 for all k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 10, 15, 1, 1, 1, 1, 35, 280, 105, 1, 1, 1, 1, 126, 5775, 15400, 945, 1, 1, 1, 1, 462, 126126, 2627625, 1401400, 10395, 1, 1, 1, 1, 1716, 2858856, 488864376, 2546168625, 190590400, 135135, 1, 1

Offset: 0

Peter Luschny, Apr 13 2023

Row n gives the leading coefficients of the set partition polynomials of type n. The sequence of these polynomial sequences starts: A097805, A048993, A156289, A291451, A291452, ...

			Array A(n, k) starts:
  [0] 1, 1,   1,       1,           1,                 1, ...
  [1] 1, 1,   1,       1,           1,                 1, ...
  [2] 1, 1,   3,      15,         105,               945, ...  A001147
  [3] 1, 1,  10,     280,       15400,           1401400, ...  A025035
  [4] 1, 1,  35,    5775,     2627625,        2546168625, ...  A025036
  [5] 1, 1, 126,  126126,   488864376,     5194672859376, ...  A025037
  [6] 1, 1, 462, 2858856, 96197645544, 11423951396577720, ...  A025038
.
Triangle A(n-k, k) starts:
  [0] 1;
  [1] 1, 1;
  [2] 1, 1,  1;
  [3] 1, 1,  1,   1;
  [4] 1, 1,  3,   1,   1;
  [5] 1, 1, 10,  15,   1, 1;
  [6] 1, 1, 35, 280, 105, 1, 1;

Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.
Antoine Chambert-Loir, Combinatorics of partitions, blog post 2024.
Alexander Karpov, Generalized knockout tournament seedings, International Journal of Computer Science in Sport, vol. 17(2), 2018.

Cf. A060540 (subarray), A370407 (antidiagonal sums, row sums).
Cf. A001147 (row 2), A025035 (row 3), A025036 (row 4), A025037 (row 5), A025038 (row 6), A025039 (row 7), A025040 (row 8), A025041 (row 9).
Cf. A088218 (column 2), A060542 (column 3), A082368 (column 4), A322252 (column 5), A057599 (main diagonal).
Cf. A097805, A048993, A156289, A291451, A291452.

A := (n, k) -> mul(binomial((j + 1)*n - 1, n - 1), j = 0..k-1):
seq(seq(A(n-k, k), k = 0..n), n = 0..9);
# Alternative, using recursion:
A := proc(n, k) local P; P := proc(n, k) option remember;
if n = 0 then return x^k*k! fi; if k = 0 then 1 else add(binomial(n*k, n*j)*
P(n,k-j)*x, j=1..k) fi end: coeff(P(n, k), x, k) / k! end:
seq(print(seq(A(n, k), k = 0..5)), n = 0..6);
# Alternative, using exponential generating function:
egf := n -> ifelse(n=0, 1, exp(x^n/n!)): ser := n -> series(egf(n), x, 8*n):
row := n -> local k; seq((n*k)!*coeff(ser(n), x, n*k), k = 0..6):
for n from 0 to 6 do [n], row(n) od;  # Peter Luschny, Aug 15 2024

A[n_, k_] := Product[Binomial[n (j + 1) - 1, n - 1], {j, 0, k - 1}]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 13 2023 *)

def Arow(n, size):
    if n == 0: return [1] * size
    return [prod(binomial((j + 1)*n - 1, n - 1) for j in range(k)) for k in range(size)]
for n in range(7): print(Arow(n, 7))
# Alternative, using exponential generating function:
def SetPolyLeadCoeff(m, n):
    x, z = var("x, z")
    if m == 0: return 1
    w = exp(2 * pi * I / m)
    o = sum(exp(z * w ** k) for k in range(m)) / m
    t = exp(x * (o - 1)).taylor(z, 0, m*n)
    p = factorial(m*n) * t.coefficient(z, m*n)
    return p.leading_coefficient(x)
for m in range(7):
    print([SetPolyLeadCoeff(m, k) for k in range(6)])

A(n, k) = (1/k!) * [x^k] P(n, k), where P(n, k) = k!*x^k if n = 0 and otherwise 1 if k = 0 and otherwise Sum_{j=1..k} binomial(n*k, n*j)*P(n, k-j)*x.

A(n, k) = (n*k)!*[x^(n*k)] exp(x^n/n!) for n >= 1. - Peter Luschny, Aug 15 2024

A264410 G.f.: Sum_{n>=0} (n^2)!/n!^(n+1) * x^n / (1-x)^(n^2+1).

Original entry on oeis.org

1, 2, 6, 299, 2630475, 5194717544512, 3708580324835714831258, 1461034854533485247412937306733067, 450538787986948219326155652866541933427447505845, 146413934927214452212855330835382443952802497537220927026261910086, 64954656894649578286072291957497800888821547620744164639686415856667115998603764278502

Offset: 0

Paul D. Hanna, Nov 26 2015

nonn

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 1714*x^3 + 63079895*x^4 + 623361815288736*x^5 +...
where
A(x) = 1/(1-x) + x/(1-x)^2 + (4!/2!^3)*x^2/(1-x)^5 + (9!/3!^4)*x^3/(1-x)^10 + (16!/4!^5)*x^4/(1-x)^17 + (25!/5!^6)*x^5/(1-x)^26 +...
Equivalently,
A(x) = 1/(1-x) + x/(1-x)^2 + 3*x^2/(1-x)^5 + 280*x^3/(1-x)^10 + 2627625*x^4/(1-x)^17 + 5194672859376*x^5/(1-x)^26 +...+ A057599(n)*x^n/(1-x)^(n^2+1) +...
Illustrate formula a(n) = Sum_{k=0..n} (1/k!)*Product_{j=0..k-1} C(n+j*k,k) for initial terms:
a(0) = 1;
a(1) = 1 + C(1,1);
a(2) = 1 + C(2,1) + C(2,2)*C(4,2)/2!;
a(3) = 1 + C(3,1) + C(3,2)*C(5,2)/2! + C(3,3)*C(6,3)*C(9,3)/3!;
a(4) = 1 + C(4,1) + C(4,2)*C(6,2)/2! + C(4,3)*C(7,3)*C(10,3)/3! + C(4,4)*C(8,4)*C(12,4)*C(16,4)/4!;
a(5) = 1 + C(5,1) + C(5,2)*C(7,2)/2! + C(5,3)*C(8,3)*C(11,3)/3! + C(5,4)*C(9,4)*C(13,4)*C(17,4)/4! + C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5)/5!; ...
which numerically equals:
a(0) = 1;
a(1) = 1 + 1 = 2;
a(2) = 1 + 2 + 1*6/2! = 6;
a(3) = 1 + 3 + 3*10/2! + 1*20*84/3! = 299;
a(4) = 1 + 4 + 6*15/2! + 4*35*120/3! + 1*70*495*1820/4! = 2630475;
a(5) = 1 + 5 + 10*21/2! + 10*56*165/3! + 5*126*715*2380/4! + 1*252*3003*15504*53130/5! = 5194717544512; ...

Paul D. Hanna, Table of n, a(n) for n = 0..30

Cf. A229050, A057599.

Table[Sum[Product[Binomial[n+j*k, k], {j, 0, k-1}]/k!, {k, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^(m+1)*x^m/(1-x +x*O(x^n))^(m^2+1)), n)}
for(n=0, 15, print1(a(n), ", "))

{a(n) = sum(k=0, n, (1/k!) * prod(j=0, k-1, binomial(n+j*k, k)) )}
for(n=0, 15, print1(a(n), ", "))

a(n) = Sum_{k=0..n} (1/k!) * Product_{j=0..k-1} binomial(n+j*k, k).

a(n) ~ exp(n - 1/12) * n^(n^2 - 3*n/2 + 1/2) / (2*Pi)^(n/2). - Vaclav Kotesovec, Dec 09 2015

A263884 a(n) = (m(n)*n)! / (n!)^(m(n)+1), where m(n) is the largest prime power <= n.

Original entry on oeis.org

1, 3, 280, 2627625, 5194672859376, 1903991899429620, 1461034854396267778567973305958400, 450538787986875167583433232345723106006796340625, 146413934927214422927834111686633731590253260933067148964500000000, 3752368324673960479843764075706478869144868251518618794695144146928706880

Offset: 1

Jonathan Sondow, Dec 19 2015

nonn

Morris and Fritze (2015) prove that a(n) is an integer.

			The largest prime power <= 6 is m(6) = 5, so a(6) = (5*6)! / (6!)^(5+1) = 30! / (6!)^6 = 1903991899429620.

Howard Carry Morris and Daniel Fritze, Problem 1948, Math. Mag., 88 (2015), 288-289.

Cf. A057599.

a(n) = A057599(n) for n a prime power.

cp's OEIS Frontend

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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A096126 a(n) is the least integer of the form (n^2)!/(n!)^k.

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A068740 Result after dividing (n^n)! as many times as possible by n!.

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

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A370364 Number of partitions of [n^2] into n sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

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Maple

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A370367 Number of partitions of [n^2] into n sets of size n having no set of consecutive numbers whose maximum (if k>n) is a multiple of n.

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Maple

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

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PARI

A361948 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} binomial((j + 1)*n - 1, n - 1) if n >= 1, and A(0, k) = 1 for all k.

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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.