A000442 -id:A000442 - cp's OEIS frontend

A001044 a(n) = (n!)^2.

1, 1, 4, 36, 576, 14400, 518400, 25401600, 1625702400, 131681894400, 13168189440000, 1593350922240000, 229442532802560000, 38775788043632640000, 7600054456551997440000, 1710012252724199424000000, 437763136697395052544000000, 126513546505547170185216000000

Offset: 0

N. J. A. Sloane and R. K. Guy

Let M_n be the symmetrical n X n matrix M_n(i,j) = 1/Max(i,j); then for n > 0 det(M_n)=1/a(n). - Benoit Cloitre, Apr 27 2002
The n-th entry of the sequence is the value of the permanent of a k X k matrix A defined as follows: k is the n-th odd number; if we concatenate the rows of A to form a vector v of length n^2, v_{i}=1 if i=1 or a multiple of 2. - Simone Severini, Feb 15 2006
a(n) = number of set partitions of {1,2,...,3n-1,3n} into blocks of size 3 in which the entries of each block mod 3 are distinct. For example, a(2) = 4 counts 123-456, 156-234, 126-345, 135-246. - David Callan, Mar 30 2007
From Emeric Deutsch, Nov 22 2007: (Start)
Number of permutations of {1,2,...,2n} with no even entry followed by a smaller entry. Example: a(2)=4 because we have 1234, 1324, 3124 and 2314.
Number of permutations of {1,2,...,2n} with n even entries that are followed by a smaller entry. Example: a(2)=4 because we have 2143, 3421, 4213 and 4321.
Number of permutations of {1,2,...,2n-1} with no even entry followed by a smaller entry. Example: a(2)=4 because we have 123, 132, 312 and 231.
Number of permutations of {1,2,...,2n-1} with n-1 odd entries followed by a smaller entry. Example: a(2)=4 because we have 132, 312, 231 and 321.
(End)
G. Leibniz in his "Ars Combinatoria" established the identity P(n)^2 = P(n-1)[P(n+1)-P(n)], where P(n) = n!. (For example, see the Burton reference.) - Mohammad K. Azarian, Mar 28 2008
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = sigma_2(gcd(i,j)) for 1 <= i,j <= n, and n>0, where sigma_2 is A001157. - Enrique Pérez Herrero, Aug 13 2011
The o.g.f. of 1/a(n) is BesselI(0,2*sqrt(x)). See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - Wolfdieter Lang, Jan 09 2012
Number of n x n x n cubes C of zeros and ones such that C(x,y,z) and C(u,v,w) can be nonzero simultaneously only if either x!=u, y!=v, or z!=w. This generalizes permutations which can be considered as n x n squares P of zeros and ones such that P(x,y) and P(u,v) can be nonzero simultaneously only if either x!=u or y!=v. - Joerg Arndt, May 28 2012
a(n) is the number of functions f:[n]->[n(n+1)/2] such that, if round(sqrt(2f(x))) = round(sqrt(2f(y))), then x=y. - Dennis P. Walsh, Nov 26 2012
From Jerrold Grossman, Jul 22 2018: (Start)
a(n) is the number of n X n 0-1 matrices whose row sums and column sums are both {1,2,...,n}.
a(n) is the number of linear arrangements of 2n blocks of n different colors, 2 of each color, such that there are an even number of blocks between each pair of blocks of the same color.
(End)
Number of ways to place n instances of a digit inside an n X n X n cube so that no two instances lie on a plane parallel to a face of the cube (see Khovanova link, Lemma 6, p. 22). - Tanya Khovanova and Wayne Zhao, Oct 17 2018
Number of permutations P of length 2n which maximize Sum_{i=1..2n} |P_i - i|. - Fang Lixing, Dec 07 2018

			Consider the square array
  1,  2,  3,  4,  5,  6, ...
  2,  4,  6,  8, 10, 12, ...
  3,  6,  9, 12, 15, 18, ...
  4,  8, 12, 16, 20, 24, ...
  5, 10, 15, 20, 25, 30, ...
  ...
then a(n) = product of n-th antidiagonal. - _Amarnath Murthy_, Apr 06 2003
a(3) = 36 since there are 36 functions f:[3]->[6] such that, if round(sqrt(2f(x))) = round(sqrt(2f(y))), then x=y. The functions, denoted by <f(1),f(2),f(3)>, are <1,2,4>, <1,2,5>, <1,2,6>, <1,3,4>, <1,3,5>, <1,3,6> and their respective permutations. - _Dennis P. Walsh_, Nov 26 2012
1 + x + 4*x^2 + 36*x^3 + 576*x^4 + 14400*x^5 + 518400*x^6 + ...

Archimedeans Problems Drive, Eureka, 22 (1959), 15.
David Burton, "The History of Mathematics", Sixth Edition, Problem 2, p. 433.
J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, No. 4/2004 (to appear).
S. M. Kerawala, The enumeration of the Latin rectangle of depth three by means of a difference equation, Bull. Calcutta Math. Soc., 33 (1941), 119-127.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.62(b).

T. D. Noe, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968.
G. S. Kazandzidis, On a Conjecture of Moessner and a General Problem, Bull. Soc. Math. Grèce, Nouvelle Série - vol. 2, fasc. 1-2, pp. 23-30, 1961.
S. M. Kerawala, The enumeration of the Latin rectangle of depth three by means of a difference equation, Bull. Calcutta Math. Soc., 33 (1941), 119-127. [Annotated scanned copy]
T. Khovanova and W. Zhao, Mathematics of a Sudo-Kurve, arXiv:1808.06713 [math.HO], 2018.
S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.
Rob Pratt (Proposer), Problem 11573, Amer. Math. Monthly, 120 (2013), 372.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Simone Severini, Home page
Index to divisibility sequences
Index entries for sequences related to factorial numbers

Cf. A000142, A000292, A084939, A084940, A084941, A084942, A084943, A084944, A020549, A046032, A048617.
First right-hand column of triangle A008955.
Cf. A134434, A134435, A000442, A134375.
Row n=2 of A225816.
Cf. A000290.
With signs, a row of A288580.

List([0..20],n->Factorial(n)^2); # Muniru A Asiru, Oct 24 2018

import Data.List (genericIndex)
a001044 n = genericIndex a001044_list n
a001044_list = 1 : zipWith (*) (tail a000290_list) a001044_list
-- Reinhard Zumkeller, Sep 05 2015

[Factorial(n)^2: n in [0..20]]; // Vincenzo Librandi, Oct 24 2018

seq((n!)^2,n=0..20); # Dennis P. Walsh, Nov 26 2012

Table[n!^2, {n, 0, 20}] (* Stefan Steinerberger, Apr 07 2006 *)
Join[{1},Table[Det[DiagonalMatrix[Range[n]^2]],{n,20}]] (* Harvey P. Dale, Mar 31 2020 *)

a(n)=n!^2 \\ Charles R Greathouse IV, Jun 15 2011

import math
for n in range(0,20): print(math.factorial(n)**2, end=', ') # Stefano Spezia, Oct 29 2018

a(n) = Integral_{x>=0} 2*BesselK(0, 2*sqrt(x))*x^n. This integral represents the n-th moment of a positive function defined on the positive half-axis. - Karol A. Penson, Oct 09 2001
a(n) ~ 2*Pi*n*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 07 2002
a(n) = polygorial(n, 4) = A000142(n)/A000079(n)*A000165(n) = (n!/2^n)*Product_{i=0..n-1} (2*i + 2) = n!*Pochhammer(1, n) = n!^2. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
a(n) = Sum_{k>=0} (-1)^k*C(n, k)^2*k!*(2*n-k)!. - Philippe Deléham, Jan 07 2004
a(n) = !n!1 = !n! = Product{i=0, 1, 2, ... .}_{0 < |n-i| <= n}(n-i) = n(n-1)(n-2)...(2)(1)(-1)(-2)...(-n+2)(-n+1)(-n) = [(-1)^n][(n!)^2]. - J. Dezert (Jean.Dezert(AT)onera.fr), Mar 21 2004
D-finite with recurrence: a(0) = 1, a(n) = n^2*a(n-1). - Arkadiusz Wesolowski, Oct 04 2011
From Sergei N. Gladkovskii, Jun 14 2012: (Start)
A(x) = Sum_{n>=0,N) a(n)*x^n = 1 + x/(U(0;N-2)-x); N >= 4; U(k)= 1 + x*(k+1)^2 - x*(k+2)^2/G(k+1); besides U(0;infinity)=x; (continued fraction).
Let B(x) = Sum_{n>=0} a(n)*x^n/((n!)*(n+s)!), then B(0) = 1/(1-x) for abs(x) < 1  and  B(1)= -1/x * log(1-x) for abs(x)< 1.
(End).
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (k+1)^2*(1 - x*G(k+1)). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = det(S(i+2,j), 1 <= i,j <= n), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013
a(n) = (2*n+1)!*2^(-4*n)*Sum_{k=0..n} (-1)^k*C(2*n+1,n-k)/(2*k+1). - Mircea Merca, Nov 12 2013
a(n) = A000290(A000142(n)). - Michel Marcus, Nov 12 2013
Sum_{n>=0} 1/a(n) = A070910 [Gradsteyn, Rzyhik 0.246.1]. - R. J. Mathar, Feb 25 2014. Corrected by Ilya Gutkovskiy, Aug 16 2016
From Ivan N. Ianakiev, Aug 16 2016: (Start)
a(n) = a(n-1) + 2*((n-1)^2)*sqrt(a(n-1)*a(n-2)) + ((n-1)^4)*a(n-2), for n > 1.
a(n) = a(n-1) - 2*(n^2 - 1)*sqrt(a(n-1)*a(n-2)) + (n^2 - 1)*a(n-2), for n > 1.
(End).
From Ilya Gutkovskiy, Aug 16 2016: (Start)
a(n) = A184877(n)*A184877(n-1).
Sum_{n>=0} (-1)^n/a(n) = BesselJ(0,2) = A091681. (End)
Sum_{n>=0} a(n)/(2*n+1)! = 2*Pi/sqrt(27). - Daniel Suteu, Feb 06 2017
a(n) = [x^n] Product_{k=1..n} (1 + k^2*x). - Vaclav Kotesovec, Feb 19 2022
a(n) = (2*n+1)! * [x^(2*n+1)] 4*arcsin(x/2)/sqrt(4-x^2). - Ira M. Gessel, Dec 10 2024

More terms from James Sellers, Sep 19 2000

More terms from Simone Severini, Feb 15 2006

A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 9, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 16, 15, 12, 7, 8, 14, 18, 20, 20, 18, 14, 8, 9, 16, 21, 24, 25, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12

Offset: 1

Marc LeBrun

Or, triangle X(n,m) = T(n-m+1,m) read by rows, in which row n gives the numbers n*1, (n-1)*2, (n-2)*3, ..., 2*(n-1), 1*n.
Radius of incircle of Pythagorean triangle with sides a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen, Aug 16 2001
A permutation of A061017. - Matthew Vandermast, Feb 28 2003
In the proof of countability of rational numbers they are arranged in a square array. a(n) = p*q where p/q is the corresponding rational number as read from the array. - Amarnath Murthy, May 29 2003
Permanent of upper right n X n corner is A000442. - Marc LeBrun, Dec 11 2003
Row 12 gives total number of partridges, turtle doves, ... and drummers drumming that you have received at the end of the Twelve Days of Christmas song. - Alonso del Arte, Jun 17 2005
Consider a particle with spin S (a half-integer) and 2S+1 quantum states |m>, m = -S,-S+1,...,S-1,S. Then the matrix element  = sqrt((S+m+1)(S-m)) of the spin-raising operator is the square-root of the triangular (tabl) element T(r,o) of this sequence in row r = 2S, and at offset o=2(S+m). T(r,o) is also the intensity || of the transition between the states |m> and |m+1>. For example, the five transitions between the 6 states of a spin S=5/2 particle have relative intensities 5,8,9,8,5. The total intensity of all spin 5/2 transitions (relative to spin 1/2) is 35, which is the tetrahedral number A000292(5). - Stanislav Sykora, May 26 2012
Sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k) = n. See A098359. - Charlie Marion, Apr 22 2013
T(n, k) is also the (k-1)-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n. - Stefano Spezia, Jul 12 2019
From Eric Lengyel, Jun 28 2023: (Start)
X(n, m+1) is the number of degrees of freedom that an m-dimensional flat geometry (point, line, plane, etc.) has when embedded in an n-dimensional Euclidean space.
X(n+1, m+1) is the number of degrees of freedom that an m-ball has when embedded in an n-dimensional Euclidean space. (End)
T(n, k) is also the average number of steps it takes a person to fall off a board of length n+k, if the person starts a random walk at k. - Ruediger Jehn, May 12 2025

			The array T starts in row n=1 with columns m>=1 as:
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
   2   4   6   8  10  12  14  16  18  20  22  24  26  28  30
   3   6   9  12  15  18  21  24  27  30  33  36  39  42  45
   4   8  12  16  20  24  28  32  36  40  44  48  52  56  60
   5  10  15  20  25  30  35  40  45  50  55  60  65  70  75
   6  12  18  24  30  36  42  48  54  60  66  72  78  84  90
   7  14  21  28  35  42  49  56  63  70  77  84  91  98 105
   8  16  24  32  40  48  56  64  72  80  88  96 104 112 120
   9  18  27  36  45  54  63  72  81  90  99 108 117 126 135
  10  20  30  40  50  60  70  80  90 100 110 120 130 140 150
The triangle X(n, m) begins
   n\m  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
   1:   1
   2:   2  2
   3:   3  4  3
   4:   4  6  6  4
   5:   5  8  9  8  5
   6:   6 10 12 12 10  6
   7:   7 12 15 16 15 12  7
   8:   8 14 18 20 20 18 14  8
   9:   9 16 21 24 25 24 21 16  9
  10:  10 18 24 28 30 30 28 24 18 10
  11:  11 20 27 32 35 36 35 32 27 20 11
  12:  12 22 30 36 40 42 42 40 36 30 22 12
  13:  13 24 33 40 45 48 49 48 45 40 33 24 13
  14:  14 26 36 44 50 54 56 56 54 50 44 36 26 14
  15:  15 28 39 48 55 60 63 64 63 60 55 48 39 28 15
  ... Formatted by _Wolfdieter Lang_, Dec 02 2014

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 46.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 5-6.

T. D. Noe, Rows n = 1..100 of triangle, flattened
Iva Kodrnja and Helena Koncul, Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N)), arXiv:2405.10747 [math.NT], 2024. See p. 10.
G. W. Leibniz, Dissertatio de arte combinatoria, 1666, Leipzig. (in Latin. This triangle appears on p. 208, page 44 of the PDF file).
Abdelkader Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9 no. 2 (1997), p. 319-335.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Main diagonal gives squares A000290. Antidiagonal sums are tetrahedral numbers A000292. See A004247 for another version.
Cf. A003989, A003990, A003056, A049581, A000442, A027424, A002260, A033638, A059895, A059896, A059897, A002620.
Cf. also A051776, A067138, A091257, A325821, A331590, A341520, A350066.

/* As triangle */ [[k*(n-k+1): k in [1..n]]: n in [1..15]]; // Vincenzo Librandi, Jul 12 2019

seq(seq(i*(n-i),i=1..n-1),n=2..10); # Robert Israel, Dec 14 2015

Table[(x + 1 - y) y, {x, 13}, {y, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)
f[n_] := Table[SeriesCoefficient[E^(x + y) (1+ x - y +x*y-y^2), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n}]; Flatten[Array[f, 11,0]] (* Stefano Spezia, Jul 12 2019 *)

PARI
```
A003991(n,k) = if(k<1 || n<1,0,k*n)
```

Rectangular array: T(n, m) = n*m, n>=1, m>= 1.
Triangle X(n, m) = T(n-m+1, m) = (n-m+1)*m.
Sum_{i=1..n} Sum_{j=1..n} a(n) = A000537(n) [Sum of first n cubes; or n-th triangular number squared.] Determinant of all n X n contiguous subarrays of A003991 is 0. - Gerald McGarvey, Sep 26 2004
G.f. as rectangular array: x*y/((1 - x)^2*(1 - y)^2).
a(n) = i*j, where i=floor((1+sqrt(8n-7))/2), j=n-i*(i-1)/2. - Hieronymus Fischer, Aug 08 2007
As an infinite lower triangular matrix equals A000012 * A002260; where A000012 = (1; 1,1; 1,1,1; ...) and A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Oct 23 2007
As a linear array, the sequence is a(n) = A002260(n)*A004736(n) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012
G.f. as linear array: (x - 3*x^2 + Sum_{k >= 0} ((k+2-x-(k+1)*x^2)*x^((k^2+3*k+4)/2)))/(1-x)^3. - Robert Israel, Dec 14 2015
E.g.f. as triangle: exp(x+y)*(1 + x - y + x*y - y^2). - Stefano Spezia, Jul 12 2019
a(n) = (1/2)*t + (n - 1/4)*t^2 - (1/4)*t^4 - n^2 + n, where t = floor(sqrt(2*n) + 1/2). - Ridouane Oudra, Nov 21 2020
a(n) = A003989(n) * A003990(n) = A059895(n) * A059896(n) = A059895(n)^2 * A059897(n). - Antti Karttunen, Dec 13 2021
T(n,k) = A002620(n+k) - A002620(n-k). - Michel Marcus, Jan 06 2023
T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x < y < z. - Clark Kimberling, Jan 22 2024
E.g.f. as rectangular array: x*y*exp(x+y). - Stefano Spezia, Jun 27 2025

More terms from Michael Somos

A204249 Permanent of the n-th principal submatrix of A003057.

Original entry on oeis.org

1, 2, 17, 336, 12052, 685080, 56658660, 6428352000, 958532774976, 181800011433600, 42745508545320000, 12203347213269273600, 4158410247782904833280, 1667267950805177583582720, 776990110000329481864608000, 416483579190482716042690560000

Offset: 0

Clark Kimberling, Jan 14 2012

nonn

I have proved that for any odd prime p we have a(p) == p (mod p^2). - Zhi-Wei Sun, Aug 30 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..36
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Cf. A000442, A003057, A085750, A085807, A278845, A278847, A278300, A278925, A278926.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i+j))):
seq(a(n), n=0..16);  # Alois P. Heinz, Nov 14 2016

f[i_, j_] := i + j;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]  (* A003057 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 15}]  (* A204249 *)

{a(n) = matpermanent(matrix(n, n, i, j, i+j))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 21 2018

From Vaclav Kotesovec, Dec 01 2016: (Start)
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A278300 = 2.455407482284127949... and c = 1.41510164826...
a(n) ~ c * d^n * n^(2*n + 1/2), where d = A278300/exp(2) = 0.332303267076220516... and c = 8.89134588451...
(End)

a(0)=1 prepended and one more term added by Alois P. Heinz, Nov 14 2016

cp's OEIS Frontend

A001044 a(n) = (n!)^2.

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A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

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A204249 Permanent of the n-th principal submatrix of A003057.

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A134375 a(n) = (n!)^4.

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A134367 a(n) = (n!)^(n-2).

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A134374 a(n) = ((2n+1)!)^2.

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A134368 a(n) = ((2n)!)^(n+1).

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