A088020 -id:A088020 - cp's OEIS frontend

A255322 a(n) = Product_{k=0..n} (k^2)!.

1, 1, 24, 8709120, 182219087869378560000, 2826438545846116156142906806150103040000000000, 1051416277636507481568264360276689674557030810000137484550133942059008000000000000000000

Offset: 0

Vaclav Kotesovec, Feb 21 2015

nonn

Partial products of A088020. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..12

Cf. A000178, A255358, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255504.
Cf. A272094, A372240, A372241.

Table[Product[(k^2)!, {k, 0, n}], {n, 0, 10}]
FoldList[Times,(Range[0,6]^2)!] (* Harvey P. Dale, Jan 30 2022 *)
Table[(n^2)!^(n+1) / Product[j^(Ceiling[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)
Table[(n^2)!^n * (n!)^2 / Product[j^(Floor[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)

{a(n) = prod(k=1, n, (k^2)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ c * n^((2*n + 1)*(2*n^2 + 2*n + 3)/6) * (2*Pi)^(n/2) / exp(5*n^3/9 + n^2/2 + n), where c = A255504 = 3.048330306522348566911920417337613015885313475... .
From Vaclav Kotesovec, Apr 23 2024: (Start)
a(n) = Product_{j=1..n^2} j^(n - ceiling(sqrt(j)) + 1).
a(n) = (n^2)!^n * (n!)^2 / Product_{j=1..n^2} j^(floor(sqrt(j))). (End)

A039622 Number of n X n Young tableaux.

Original entry on oeis.org

1, 1, 2, 42, 24024, 701149020, 1671643033734960, 475073684264389879228560, 22081374992701950398847674830857600, 220381378415074546123953914908618547085974856000, 599868742615440724911356453304513631101279740967209774643120000

Offset: 0

Floor van Lamoen

Number of arrangements of 1,2,...,n^2 in an n X n array such that each row and each column is increasing. The problem for a 5 X 5 array was recently posed and solved in the College Mathematics Journal. See the links.
This is the factor g_n that appears in a conjectured formula for 2n-th moment of the Riemann zeta function on the critical line. (See Conrey articles.) - Michael Somos, Apr 15 2003 [Comment revised by N. J. A. Sloane, Jun 21 2016]
Number of linear extensions of the n X n lattice. - Mitch Harris, Dec 27 2005

			Using the hook length formula, a(4) = (16)!/(7*6^2*5^3*4^4*3^3*2^2) = 24024.

M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 284.

Alois P. Heinz, Table of n, a(n) for n = 0..30
P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
Joerg Arndt, The a(3)=42 3 X 3 Young tableaux
J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 349.
J. B. Conrey, Review of H. Iwaniec, "Lectures on the Riemann Zeta Function" (AMS, 2014), Bull. Amer. Math. Soc., 53 (No. 3, 2016), 507-512.
P.-O. Dehaye, Combinatorics of the lower order terms in the moment conjectures: the Riemann zeta function, arXiv preprint arXiv:1201.4478 [math.NT], 2012.
J. S. Frame, G. de B. Robinson and R. M. Thrall, The hook graphs of a symmetric group, Canad. J. Math. 6 (1954), pp. 316-324.
Curtis Greene and Brady Haran, Shapes and Hook Numbers, Numberphile video (2016)
Curtis Greene and Brady Haran, Shapes and Hook Numbers (extra footage) (2016)
Zachary Hamaker and Eric Marberg, Atoms for signed permutations, arXiv:1802.09805 [math.CO], 2018.
Alejandro H. Morales, I. Pak, and G. Panova, Why is pi < 2 phi?, Preprint, 2016; The American Mathematical Monthly, Volume 125, 2018 - Issue 8.
Alan H. Rapoport (proposer), Solution to Problem 639: A Square Young Tableau, College Mathematics Journal, Vol. 30 (1999), no. 5, pp. 410-411.
Index entries for sequences related to Young tableaux.

Main diagonal of A060854.
Also a(2)=A000108(2), a(3)=A005789(3), a(4)=A005790(4), a(5)=A005791(5).
Cf. A000178, A000984, A079478, A088020, A107252, A107254, A127223, A153452.

A039622:= func< n | n eq 0 select 1 else Factorial(n^2)*(&*[Factorial(j)/Factorial(n+j): j in [0..n-1]]) >;
[A039622(n): n in [0..12]]; // G. C. Greubel, Apr 21 2021

a:= n-> (n^2)! *mul(k!/(n+k)!, k=0..n-1):
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 10 2012

a[n_]:= (n^2)!*Product[ k!/(n+k)!, {k, 0, n-1}]; Table[ a[n], {n, 0, 12}] (* Jean-François Alcover, Dec 06 2011, after Pari *)

PARI
```
a(n)=(n^2)!*prod(k=0,n-1,k!/(n+k)!)
```

def A039622(n): return factorial(n^2)*product( factorial(j)/factorial(n+j) for j in (0..n-1))
[A039622(n) for n in (0..12)] # G. C. Greubel, Apr 21 2021

a(n) = (n^2)! / Product_{k=1..2n-1} k^(n - |n-k|).
a(n) = 0!*1!*...*(k-1)! *(k*n)! / ( n!*(n+1)!*...*(n+k-1)! ) for k=n.
a(n) = A088020(n)/A107254(n) = A088020(n)*A000984(n)/A079478(n). - Henry Bottomley, May 14 2005
a(n) = A153452(prime(n)^n). - Naohiro Nomoto, Jan 01 2009
a(n) ~ sqrt(Pi) * n^(n^2+11/12) * exp(n^2/2+1/12) / (A * 2^(2*n^2-7/12)), where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 10 2015
From Peter Luschny, May 20 2019: (Start)
a(n) = (G(1+n)*G(2+n)^(2-n)*(n^2)!*(G(3+n)/Gamma(2+n))^(n-1))/(G(1+2*n)*n!) where G(x) is the Barnes G function.
a(n) = A127223(n) / A107252(n). (End)
a(n) = (Gamma(n^2 +1)/Gamma(n+1))*(G(n+1)*G(n+2)/G(2*n+1)), where G(n) is the Barnes G-function. - G. C. Greubel, Apr 21 2021
a(n+2) = (n+2) * A060856(n+1) for n >= 0. - Tom Copeland, May 30 2022

A089484 Number of positions of the 15-puzzle at a distance of n moves from an initial state with the empty square in one of the corners, in the single-tile metric.

Original entry on oeis.org

1, 2, 4, 10, 24, 54, 107, 212, 446, 946, 1948, 3938, 7808, 15544, 30821, 60842, 119000, 231844, 447342, 859744, 1637383, 3098270, 5802411, 10783780, 19826318, 36142146, 65135623, 116238056, 204900019, 357071928, 613926161, 1042022040

Offset: 0

Hugo Pfoertner, Nov 25 2003

The single-tile metric counts moves of individual tiles as 1 move. Moving multiple tiles at once counts as more than one move, e.g. simultaneously sliding 3 tiles along a row or column counts as 3 moves.

The last term is a(80). The total number of possible configurations of an m X m sliding block puzzle is (m*m)!/2 = A088020(4)/2, therefore, Sum_i (i=0..80) a(i) = 16!/2 = 10461394944000.

See A087725.

Herman Jamke, Table of n, a(n) for n = 0..80 (complete sequence)
Herbert Kociemba, 15-Puzzle Optimal Solver
R. E. Korf and P. Schultze, Large-Scale Parallel Breadth-First Search
Hugo Pfoertner, Configuration counts for n*n sliding block puzzles.

Cf. A087725, A089473, A090031, A090032, A090164, A090165, A088020.

# alst(), moves(), swap() in A089473
start, shape = "-123456789ABCDEF", (4, 4)
alst(start, shape, v=True) # Michael S. Branicky, Dec 31 2020

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006

Name edited by Ben Whitmore, Aug 02 2024

A088021 a(n) = (n^2)!/(n!)^2.

Original entry on oeis.org

1, 1, 6, 10080, 36324288000, 1077167364120207360000, 717579719887926731226850787328000000, 23946596436219275985459662514223331478629410406400000000

Offset: 0

Hugo Pfoertner, Sep 18 2003

nonn

Based on an observation of Hugo Pfoertner, W. Edwin Clark conjectured and Xiang-dong Hou proved that (n^2)!/(n!)^2 gives the number of distinct determinants of the generic n X n matrix whose entries are n^2 different indeterminates under all (n^2)! permutations of the entries.
Using J. T. Schwarz's Sparse Zeros Lemma this implies that for any positive integer n there is an n X n matrix A with positive integer entries such that the set of determinant values obtained from A by permuting the elements of A is (n^2)!/(n!)^2.
Moreover, for any entries, no larger number of determinants can be obtained. In fact, by the Sparse Zeros Lemma one can select the entries of A from any sufficiently large subset of real numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..21

Cf. A001044, A088020.

[Factorial(n^2)/Factorial(n)^2: n in [0..10]]; // Vincenzo Librandi, May 31 2011

a(n) = A088020(n)/A001044(n).

A090031 Number of configurations of the 5 X 5 variant of sliding block 15-puzzle ("24-puzzle") that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

Original entry on oeis.org

1, 2, 4, 10, 26, 64, 159, 366, 862, 1904, 4538, 10238, 24098, 53186, 123435, 268416, 616374, 1326882, 3021126, 6438828, 14524718, 30633586, 68513713, 143106496, 317305688, 656178756, 1442068376, 2951523620, 6427133737, 13014920506, 28070588413, 56212979470, 120030667717

Offset: 0

Hugo Pfoertner, Nov 25 2003

The 15-block puzzle is often referred to (incorrectly) as Sam Loyd's 15-Puzzle.
Sum of sequence terms = A088020(5)/2.
152 <= (number of last sequence term) <= 205 (see A087725 and cube archives link for current status).  - Hugo Pfoertner, Feb 12 2020

See A087725 for references.

Robert Clausecker, term generator puzzledist.c
Robert Clausecker, The Quality of Heuristic Functions for IDA*, Zuse Institute Berlin (2020).
Hugo Pfoertner, Configuration counts for n*n sliding block puzzles.
Tomas Rokicki, comment in Twenty-Four puzzle, some observations
Ben Whitmore in the Cube Forum, 5x5 sliding puzzle can be solved in 205 moves, with updates by Johan de Ruiter claiming 182 moves.

Cf. A087725, A088020, A089473, A089484, A090032.

C
```
/* See Clausecker link. */
```
Fortran
```
! See link in A089473.
```

# alst(), moves(), swap() in A089473
start, shape = "-123456789ABCDEFGHIJKLMNO", (5, 5)
alst(start, shape, v=True) # Michael S. Branicky, Dec 31 2020

More terms from Tomas Rokicki, Aug 09 2011
a(28)-a(30) from Robert Clausecker, Jan 29 2018
a(31)-a(32) from Robert Clausecker, Sep 14 2020

A107254 a(n) = SF(2n-1)/SF(n-1)^2 where SF = A000178.

Original entry on oeis.org

1, 1, 12, 8640, 870912000, 22122558259200000, 222531556847250309120000000, 1280394777025250130271722799104000000000, 5746332926632566442385615219551212618645504000000000000

Offset: 0

Henry Bottomley, May 14 2005

nonn

Inverse product of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n). - Alexander Adamchuk, Apr 12 2006

The n X n matrix with A(i,j) = 1/(i+j-1)! (i,j = 1..n) has determinant (-1)^floor(n/2)/a(n). - Mikhail Lavrov, Nov 01 2022

			a(3) = 1!*2!*3!*4!*5!/(1!*2!*1!*2!) = 34560/4 = 8640.
n = 2: HilbertMatrix[n,n]
  1/1 1/2
  1/2 1/3
so a(2) = 1 / (1 * 1/2 * 1/2 * 1/3) = 12.
The n X n Hilbert matrix begins:
  1/1 1/2 1/3 1/4  1/5  1/6  1/7  1/8 ...
  1/2 1/3 1/4 1/5  1/6  1/7  1/8  1/9 ...
  1/3 1/4 1/5 1/6  1/7  1/8  1/9 1/10 ...
  1/4 1/5 1/6 1/7  1/8  1/9 1/10 1/11 ...
  1/5 1/6 1/7 1/8  1/9 1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...

Alois P. Heinz, Table of n, a(n) for n = 0..20
Mathematics Stack Exchange, Determinant of a matrix involving factorials.
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A000178, A002457, A005249, A098118.

A107254:= func< n | n eq 0 select 1 else (&*[Factorial(n+j)/Factorial(j): j in [0..n-1]]) >;
[A107254(n): n in [0..12]]; // G. C. Greubel, Apr 21 2021

a:= n-> mul((n+i)!/i!, i=0..n-1):
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 23 2012

Table[Product[(i+j-1),{i,1,n},{j,1,n}], {n,1,10}] (* Alexander Adamchuk, Apr 12 2006 *)
Table[n!*BarnesG[2n+1]/(BarnesG[n+2]*BarnesG[n+1]), {n,0,12}] (* G. C. Greubel, Apr 21 2021 *)

a = lambda n: prod(rising_factorial(k,n) for k in (1..n))
print([a(n) for n in (0..10)]) # Peter Luschny, Nov 29 2015

a(n) = n!*(n+1)!*(n+2)!*...*(2n-1)!/(0!*1!*2!*3!*...*(n-1)!) = A000178(2n-1)/A000178(n-1)^2 = A079478(n)/A000984(n) = A079478(n-1)*A009445(n-1) = A107252(n)*A000142(n) = A088020(n)/A039622(n).
a(n) = 1/Product_{j=1..n} ( Product_{i=1..n} 1/(i+j-1) ). - Alexander Adamchuk, Apr 12 2006
a(n) = 2^(n*(n-1)) * A136411(n) for n > 0 . - Robert Coquereaux, Apr 06 2013
a(n) = A136411(n) * A053763(n) for n > 0. [Following remark from Robert Coquereaux] - M. F. Hasler, Apr 06 2013
a(n) ~ A * 2^(2*n^2-1/12) * n^(n^2+1/12) / exp(3*n^2/2+1/12), where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 10 2015
a(n) = Product_{k=1..n} rf(k,n) where rf denotes the rising factorial. - Peter Luschny, Nov 29 2015
a(n) = (n! * G(2*n+1))/(G(n+1)*G(n+2)), where G(n) is the Barnes G - function. - G. C. Greubel, Apr 21 2021

A179268 Product of numbers between and including n and n^2.

Original entry on oeis.org

1, 24, 181440, 3487131648000, 646300418472124416000000, 3099944389915843478899995401256960000000, 844835922269816056767016893501799134566045599137792000000000

Offset: 1

Reinhard Zumkeller, Jul 06 2010

nonn

a(n) = Product_{k=n..n^2} k;

a(n) = A088020(n)/A000142(n-1).

			a(2) = 2*3*4 = 24;
a(3) = 3*4*5*6*7*8*9 = 181440.

Vincenzo Librandi, Table of n, a(n) for n = 1..23

Cf. A126804, A027441.

[Factorial(n^2) / Factorial(n-1): n in [1..10]]; // Vincenzo Librandi, May 31 2011

Table[Times@@Range[n,n^2],{n,10}] (* Harvey P. Dale, Sep 16 2020 *)

a(n) = (n^2)! / (n-1)!.

Definition clarified by Harvey P. Dale, Sep 16 2020

A221976 The number of n X n matrices with zero determinant and with entries a permutation of [1,2,..,n^2].

Original entry on oeis.org

0, 0, 2736, 8290316160

Offset: 1

R. J. Mathar, May 12 2013

This counts a subset of all (n^2)! = A088020(n) matrices which contain elements which are a permutation of [n^2]. The range of determinants is characterized in A085000, and the size of the set of different determinants in A088217.

Because any combination of row and column permutation of matrices with distinct elements generates (n!)^2 = A001044(n) different matrices, and because these restricted permutations leave the (absolute value of) the determinant constant, a(n) is a multiple of A001044(n). This factor does not yet take into account that matrix transpositions also maintain the values of determinants (and which never can be achieved by row or column permutation).

a(n) = A136609(n)*A001044(n).

A362187 a(n) = (n^2 - n)!.

Original entry on oeis.org

1, 1, 2, 720, 479001600, 2432902008176640000, 265252859812191058636308480000000, 1405006117752879898543142606244511569936384000000000, 710998587804863451854045647463724949736497978881168458687447040000000000000

Offset: 0

Stefano Spezia, Apr 10 2023

The next term has 104 digits.

For n > 0, a(n) is the number of n X n matrices using all the integers from 1 to n^2 and having the main diagonal given.

Winston de Greef, Table of n, a(n) for n = 0..21

Cf. A000142, A000290, A002378, A088020.
Cf. A039622, A179268, A272094, A272095, A272164.
Cf. A362209.

Mathematica
```
a[n_]:=(n^2-n)!; Array[a,9,0]
```

a(n) = (n^2 - n)*a(n-1) for n > 1.

a(n) = A000142(A002378(n-1)) for n > 0.

A364203 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the integers from 1 to n^2.

Original entry on oeis.org

1, 0, 24, 0, 2736, 360144

Offset: 1

Stefano Spezia, Jul 13 2023

			The triangle begins:
  1;
  0,   24;
  0, 2736, 360144;
  ...

Cf. A085000 (maximal determinant), A088020 (row sums), A350565 (minimal permanent), A350566 (maximal permanent), A364206 (right diagonal).

Cf. A364226 (with prime numbers).

cp's OEIS Frontend

A255322 a(n) = Product_{k=0..n} (k^2)!.

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A039622 Number of n X n Young tableaux.

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A089484 Number of positions of the 15-puzzle at a distance of n moves from an initial state with the empty square in one of the corners, in the single-tile metric.

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A088021 a(n) = (n^2)!/(n!)^2.

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A090031 Number of configurations of the 5 X 5 variant of sliding block 15-puzzle ("24-puzzle") that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

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A107254 a(n) = SF(2n-1)/SF(n-1)^2 where SF = A000178.

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Magma

Maple

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Sage

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A179268 Product of numbers between and including n and n^2.

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