A184877 -id:A184877 - 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

A001818 Squares of double factorials: (135*...(2n-1))^2 = ((2n-1)!!)^2.

Original entry on oeis.org

1, 1, 9, 225, 11025, 893025, 108056025, 18261468225, 4108830350625, 1187451971330625, 428670161650355625, 189043541287806830625, 100004033341249813400625, 62502520838281133375390625, 45564337691106946230659765625, 38319607998220941779984862890625

Offset: 0

N. J. A. Sloane

Number of permutations in S_{2n} in which all cycles have even length (cf. A087137).
Also number of permutations in S_{2n} in which all cycles have odd length. - Vladeta Jovovic, Aug 10 2007
a(n) is the sum over all multinomials M2(2*n,k), k from {1..p(2*n)} restricted to partitions with only even parts. p(2*n)= A000041(2*n) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n,k). - Wolfdieter Lang, Aug 07 2007
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture 1: For any primitive 2n-th root zeta of unity, the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} coincides with a(n) = ((2n-1)!!)^2, where m(j,k) is (1+zeta^(j-k))/(1-zeta^(j-k)) if j is not equal to k, and 1 otherwise.
The determinant of [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n-1)!!)^2/(2n-1) by Han Wang and Zhi-Wei Sun in 2022.
Conjecture 2: Let p be an odd prime. Then the permanent of (p-1) X (p-1) matrix [f(j,k)]_{j,k=1..p-1} is congruent to a((p-1)/2) = ((p-2)!!)^2 modulo p^2, where f(j,k) is (j+k)/(j-k) if j is not equal to k, and f(j,k) = 1 otherwise. (End)

			Multinomial representation for a(2): partitions of 2*2=4 with even parts only: (4) with position k=1, (2^2) with k=3; M2(4,1)= 6 and M2(4,3)= 3, adding up to a(2)=9.
G.f. = 1 + x + 9*x^2 + 225*x^3 + 11025*x^4 + 893025*x^5 + 108056025*x^6 + ...

John 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).
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.34(c).

T. D. Noe, Table of n, a(n) for n = 0..50
Ron M. Adin, Pál Hegedűs, and Yuval Roichman, Descent set distribution for permutations with cycles of only odd or only even lengths, arXiv:2502.03507 [math.CO], 2025. See p. 2.
David Callan and Emeric Deutsch, The Run Transform, arXiv preprint arXiv:1112.3639 [math.CO], 2011.
William Y. C. Chen and Elena L. Wang, r-Enriched Permutations and an Inequality of Bóna-McLennan-White, arXiv:2502.04136 [math.CO], 2025. See p. 4.
Harry Crane and Peter McCullagh, Reversible Markov structures on divisible set partitions, Journal of Applied Probability, Vol. 52, No. 3 (2015), pp. 622-635.
Muhammad Adam Dombrowski and Gregory Dresden, Areas Between Cosines, arXiv:2404.17694 [math.CO], 2024. See p. 11.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 6.
IBM, "Ponder This" puzzle for June 2009. [From _Vladeta Jovovic_, Jul 26 2009]
John Riordan and N. J. A. Sloane, Correspondence, 1974.
Terence Tao, A differentiation identity.
Han Wang and Zhi-Wei Sun, Proof of a conjecture involving derangements and roots of unity, arXiv:2206.02589 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Struve function.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
Index to divisibility sequences.

Cf. A001147, A002454, A111595, A197037.
Bisection of A012248.
Right-hand column 1 in triangle A008956.

DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((2*n-1))^2: n in [0..20] ]; // Vincenzo Librandi, Jul 21 2017

a := proc(m) local k; 4^m*mul((-1)^k*(k-m-1/2),k=1..2*m) end; # Peter Luschny, Jun 01 2009

FoldList[Times,1,Range[1,25,2]]^2 (* or *) Join[{1},(Range[1,29,2]!!)^2] (* Harvey P. Dale, Jun 06 2011, Apr 10 2012 *)
Table[((2 n - 1)!!)^2, {n, 0, 30}] (* Vincenzo Librandi, Jul 21 2017 *)

PARI
```
a(n)=((2*n)!/(n!*2^n))^2
```

{a(n) = if( n<0, 1 / a(-n), sqr((2*n)! / (n! * 2^n)))}; /* Michael Somos, Jan 06 2017 */

a(n) = A001147(n)^2.
a(n) = A111595(2*n, 0).
a(n) = (2*n-1)!*Sum_{k=0..n-1} binomial(2*k,k)/4^k, n >= 1. - Wolfdieter Lang, Aug 23 2005
arcsinh(x) = Sum_{n>=1} (-1)^(n-1)*a(n)*x^(2*n-1)/(2*n-1)!. - James R. Buddenhagen, Mar 24 2009
From Karol A. Penson, Oct 21 2009: (Start)
G.f.: Sum_{n>=0} a(n)*x^n/(n!)^2 = 2*EllipticK(2*sqrt(x))/Pi.
Asymptotically: a(n) = (2/((exp(-1/2))^2*(exp(1/2))^2)-1/(6*(exp(-1/2))^2*(exp(1/2))^2*n)+1/(144*(exp(-1/2))^2*(exp(1/2))^2*n^2)+O(1/n^3))*(2^n)^2/(((1/n)^n)^2*(exp(n))^2), n->infinity.
Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation:
a(n) = Integral_{x>=0} x^n*BesselK(0,sqrt(x))/(Pi*sqrt(x)).
This solution is unique.
(End)
D-finite with recurrence: a(0) = 1, a(n) = (2*n-1)^2*a(n-1), n > 0.
a(n) ~ 2*2^(2*n)*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: 1/sqrt(1-x^2) = Sum_{n >= 0} a(n)*x^(2*n)/(2*n)!. Also arcsin(x) = Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)!. - Michael Somos, Jul 03 2002
(-1)^n*a(n) is the coefficient of x^0 in prod(k=1, 2*n, x+2*k-2*n-1). - Benoit Cloitre and Michael Somos, Nov 22 2002
-arccos(x) + Pi/2 = x + x^3/3! + 9*x^5/5! + 225*x^7/7! + 11205*x^9/9! + ... - Tom Copeland, Oct 23 2008
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) =  1 - (4*k^2+4*k+1)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = det(V(i+1,j), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices. - Mircea Merca, Apr 04 2013
a(n) = (1+x^2)^(n+1/2) * (d/dx)^(2*n) (1+x^2)^(n-1/2).  See Tao link. - Robert Israel, Jun 04 2015
a(n) = 4^n * gamma(n + 1/2)^2 / Pi. - Daniel Suteu, Jan 06 2017
0 = a(n)*(+384*a(n+2) - 60*a(n+3) + a(n+4)) + a(n+1)*(-36*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) and a(n) = 1/a(-n) for all n in Z. - Michael Somos, Jan 06 2017
From Robert FERREOL, Jul 30 2020: (Start)
a(n) = ((2*n)!/4^n)*binomial(2*n,n).
a(n) = (2*n-1)!*Sum_{k=0..n-1} a(k)/(2*k)!, n >= 1.
a(n) = A184877(2*n-1) for n>=1. (End)
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + L_0(1)*Pi/2, where L is the modified Struve function (see A197037).
Sum_{n>=0} (-1)^n/a(n) = 1 - H_0(1)*Pi/2, where H is the Struve function. (End)

Incorrect formula deleted by N. J. A. Sloane, Jul 03 2009

A288580 Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-ki| <= n} (n-ki), with n >= 0, k >= 1.

Original entry on oeis.org

1, 1, -1, 1, -1, 4, 1, 1, -4, -36, 1, 1, -2, 9, 576, 1, 1, -4, -9, 64, -14400, 1, 1, 2, -3, -8, -225, 518400, 1, 1, 2, -6, -16, 40, -2304, -25401600, 1, 1, 2, -9, -4, -15, 324, 11025, 1625702400, 1, 1, 2, 3, -8, -25, 144, 280, 147456, -131681894400, 1, 1, 2, 3, -12, -5, -24, 105, -2240, -893025, 13168189440000

Offset: 0

N. J. A. Sloane, Jul 03 2017

			Array begins:
1, -1, 4, -36, 576, -14400, 518400, -25401600, 1625702400, -131681894400,  ...
1, -1, -4, 9, 64, -225, -2304, 11025, 147456, -893025, -14745600, 108056025, ...
1, 1, -2, -9, -8, 40, 324, 280, -2240, -26244, -22400, 246400, 3779136, ...
1, 1, -4, -3, -16, -15, 144, 105, 1024, 945, -14400, -10395, -147456, ...
1, 1, 2, -6, -4, -25, -24, -42, 336, 216, 2500, 2376, 4032, ...
1, 1, 2, -9, -8, -5, -36, -35, -64, 729, 640, 385, 5184, ...
1, 1, 2, 3, -12, -10, -6, -49, -48, -90, -120, 1320, 1080, ...
1, 1, 2, 3, -16, -15, -12, -7, -64, -63, -120, -165, 2304, ...
1, 1, 2, 3, 4, -20, -18, -14, -8, -81, -80, -154, -216, ...
1, 1, 2, 3, 4, -25, -24, -21, -16, -9, -100, -99, -192, ...
...

F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.

Rows k=1 through 9 are signed A001044 or A092396, signed A184877 or A092397, A092398, A092399, A092971, A092972, A092973, A092974,

T:=proc(n,k)  local i,p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
scan1:=proc(a,M1) local lis,n,k; lis:=[]; for n from 1 to M1 do for k from 0 to n-1 do
lis:=[op(lis),a(k,n-k)]; od: od: lis; end:
scan1(T,12);

T[n_, k_] := Module[{i, p = 1}, For[i = 0, i <= Floor[2n/k], i++, If[n - k i != 0, p *= (n - k i)]]; p]; T[_, 0] = 1;
Table[T[k, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 05 2020, after Maple *)

A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)(x-(n-2)^2)(x-(n-4)^2)...(x-(1 or 2)^2), highest powers first.

Original entry on oeis.org

1, 1, -1, 1, -4, 1, -10, 9, 1, -20, 64, 1, -35, 259, -225, 1, -56, 784, -2304, 1, -84, 1974, -12916, 11025, 1, -120, 4368, -52480, 147456, 1, -165, 8778, -172810, 1057221, -893025, 1, -220, 16368, -489280, 5395456, -14745600, 1, -286, 28743, -1234948, 21967231, -128816766, 108056025, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400

Offset: 0

N. J. A. Sloane, Feb 01 2011

These are scaled versions of the central factorial numbers in A008955 and A008956.
Even-indexed rows give A182867, odd-indexed rows give A008956.
A121408 is an unsigned and aerated version of the row reverse of this triangle. - Peter Bala, Aug 29 2012

			Triangle begins:
1
1, -1
1, -4
1, -10, 9
1, -20, 64
1, -35, 259, -225
1, -56, 784, -2304
1, -84, 1974, -12916, 11025
1, -120, 4368, -52480, 147456
1, -165, 8778, -172810, 1057221, -893025
1, -220, 16368, -489280, 5395456, -14745600
...
E.g. for n=5 Q(5) = (x-1^2)*(x-3^2)*(x-5^2) = x^3-35*x^2+259*x-225.

Even-indexed rows give A182867, odd-indexed rows give A008956.
Column 1,4,10,20, ... is A000292. The next two columns give A181888, A184878. The last diagonal is A184877.
Cf. A008955, A008956. A121408.

Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
for n from 0 to 12 do
t1:=eval(Q(n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od:

For n even, let Q(n) = Product_{i=1..n/2} (x - (2*i)^2) and for n odd let Q(n) = Product_{i=0..(n-1)/2} (x - (2i+1)^2). n-th row of triangle gives coefficients in expansion of Q(n).

cp's OEIS Frontend

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

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A001818 Squares of double factorials: (1*3*5*...*(2n-1))^2 = ((2*n-1)!!)^2.

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A288580 Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-k*i| <= n} (n-k*i), with n >= 0, k >= 1.

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A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)*(x-(n-2)^2)*(x-(n-4)^2)*...*(x-(1 or 2)^2), highest powers first.

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A290770 a(n) = Product_{k=1..n} k^(2*k).

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A330506 Expansion of e.g.f. Sum_{k>=1} arcsin(x^k).

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A001818 Squares of double factorials: (135*...(2n-1))^2 = ((2n-1)!!)^2.

A288580 Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-ki| <= n} (n-ki), with n >= 0, k >= 1.

A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)(x-(n-2)^2)(x-(n-4)^2)...(x-(1 or 2)^2), highest powers first.