A001044 -id:A001044 - cp's OEIS frontend

A134434 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k even entries that are followed by a smaller entry (n>=0, k>=0).

1, 1, 1, 1, 4, 2, 4, 16, 4, 36, 72, 12, 36, 324, 324, 36, 576, 2592, 1728, 144, 576, 9216, 20736, 9216, 576, 14400, 115200, 172800, 57600, 2880, 14400, 360000, 1440000, 1440000, 360000, 14400, 518400, 6480000, 17280000, 12960000, 2592000, 86400

Offset: 0

Emeric Deutsch, Nov 22 2007

Row n has 1+floor(n/2) entries. T(2n-1,0) = T(2n,0) = T(2n,n) = (n!)^2 = A001044(n).

This descent statistic is equidistributed on the symmetric group S_n with a multiplicative 2-excedance statistic - see A136715 for details. - Peter Bala, Jan 23 2008

			T(4,2) = 4 because we have 2143, 4213, 3421 and 4321.
Triangle starts:
   1;
   1;
   1,   1;
   4,   2;
   4,  16,   4;
  36,  72,  12;
  36, 324, 324, 36;
  ...

S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.

Bisection of column k=0 gives A001044 (even part).
Row sums give A000142.
Cf. A134435, A136715.

R[0]:=1:R[1]:=1: R[2]:=1+t: for n to 5 do R[2*n+1]:=sort(expand((1-t)* (diff(R[2*n], t))+(2*n+1)*R[2*n])): R[2*n+2]:=sort(expand(t*(1-t)*(diff(R[2*n+1], t))+(1+(2*n+1)*t)*R[2*n+1])) end do: for n from 0 to 11 do seq(coeff(R[n], t, j), j=0..floor((1/2)*n)); end do; # yields sequence in triangular form

T[n_,k_]:=If[EvenQ[n],Floor[(n/2)!Binomial[n/2,k]]^2, Floor[((n+1)/2)!Binomial[(n-1)/2,k]]^2/(k+1)]; Table[T[n,k],{n,11},{k,0,Floor[n/2]}]//Flatten (* Stefano Spezia, Jul 12 2024 *)

T(2n,k) = [n!*C(n,k)]^2; T(2n+1,k) = [(n+1)!*C(n,k)]^2/(k+1). See the Kitaev & Remmel reference for recurrence relations (Sec. 3).

T(0,0)=1 prepended by Alois P. Heinz, Jul 12 2024

A166602 Numbers k such that Sum_{i=1..k} i^2 divides Product_{i=1..k} i^2.

Original entry on oeis.org

1, 7, 13, 17, 19, 24, 25, 27, 31, 32, 34, 37, 38, 43, 45, 47, 49, 55, 57, 59, 61, 62, 64, 67, 71, 73, 76, 77, 79, 80, 84, 85, 87, 91, 92, 93, 94, 97, 101, 103, 104, 107, 109, 110, 115, 117, 118, 121, 122, 123, 124, 127, 129, 132, 133, 137, 139, 142, 143, 144, 145, 147

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

Product_{i=1..k} i^2 = (k!)^2 and Sum_{i=1..k} i^2 = k*(k+1)*(2*k+1)/6. - J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

			a(2) = A125314(2) = 7.

J. Mulder, Table of n, a(n) for a(n) below 20000

Cf. A000330, A001044, A125294, A125314, A060462, A166604, A166605, A166606, A166607, A166608, A166609, A166610, A334735.

Cf. A067656. - R. J. Mathar, Oct 23 2009

q:= k-> is(irem(k!^2, k*(k+1)*(2*k+1)/6)=0):
select(q, [$1..200])[];  # Alois P. Heinz, May 09 2020

Cases[Range[2, 5000], k_ /; Divisible[Factorial[k - 1]^2, 1/6 (-1 + k) k (-1 + 2 k)]] - 1 (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

isok(k) = ((k!)^2 % (k*(k+1)*(2*k+1)/6)) == 0; \\ Michel Marcus, May 09 2020

Terms below 5000 by J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

More terms copied from the b-file by R. J. Mathar, Feb 14 2010

A055546 a(n) = (-1)^(n+1) * 2^n * n!^2.

Original entry on oeis.org

-1, 2, -16, 288, -9216, 460800, -33177600, 3251404800, -416179814400, 67421129932800, -13484225986560000, 3263182688747520000, -939796614359285760000, 317651255653438586880000, -124519292216147926056960000, 56033681497266566725632000000

Offset: 0

Eric W. Weisstein

sign

Coefficient of the Cayley-Menger determinant of order n.
A roller coaster has n rows of seats, each of which has room for two people.  |a(n)| is the number of ways n men and n women can be seated with a man and a woman in each row. - Geoffrey Critzer, Dec 17 2011
The o.g.f. of 1/a(n) is -BesselI(0,i*sqrt(2*x)), with i the imaginary unit. See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - Wolfdieter Lang, Jan 10 2012
|a(n)|/2 is the number of integers k such that the digits of k and 2*k, written in base 2*n, are permutations of 0, 1, ..., 2*n-1. - Yifan Xie, Apr 12 2025

Andrew Howroyd, Table of n, a(n) for n = 0..100
Paul C. Kainen, Construction numbers: How to build a graph?, arXiv:2302.13186 [math.CO], 2023.
Usman A. Khan, Soummya Kar and Jose M. F. Moura, A novel geometric approach towards a linear theory for sensor localization, 2013.
Alan L. Mackay, On the regular heptagon, J. Math. Chemistry, vol. 21, 1997, 197-209.
Eric Weisstein's World of Mathematics, Cayley-Menger Determinant.

Cf. A000079, A001044, A019669, A334381, A334383.

Row of A340591 (in absolute values).

Mathematica
```
Table[(-1)^(n+1)2^n n!^2, {n, 0, 20}]
```

a(n)={(-1)^(n+1) * 2^n * n!^2} \\ Andrew Howroyd, Nov 07 2019

E.g.f.: -arcsinh(x/sqrt(2))^2. - Vladeta Jovovic, Aug 30 2004
Sum_{n>=0} |a(n)|/(2*n+1)! = Pi/2. - Daniel Suteu, Feb 06 2017
a(n) = (-1)^(n+1) * A000079(n) * A001044(n). - Terry D. Grant, May 21 2017
From Amiram Eldar, Nov 18 2020: (Start)
Sum_{n>=0} 1/a(n) = (-1) * A334383.
Sum_{n>=0} (-1)^(n+1)/a(n) = A334381. (End)

Terms a(14) and beyond from Andrew Howroyd, Nov 07 2019

A134372 a(n) = ((2n)!)^2.

Original entry on oeis.org

1, 4, 576, 518400, 1625702400, 13168189440000, 229442532802560000, 7600054456551997440000, 437763136697395052544000000, 40990389067797283140009984000000, 5919012181389927685417441689600000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A010050, A134366, A134367, A134368, A134369, A134371, A134374, A134375, A334379, A334632.

Table[((2n)!)^(2), {n, 0, 10}]
((2*Range[0,20])!)^2 (* Harvey P. Dale, Jul 14 2011 *)

a(n) = ((2*n)!)^2; \\ Michel Marcus, Nov 16 2020

From Amiram Eldar, Nov 16 2020: (Start)
Sum_{n>=0} 1/a(n) = A334379.
Sum_{n>=0} (-1)^n/a(n) = A334632. (End)

A269944 Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 5, 1, 0, 36, 49, 14, 1, 0, 576, 820, 273, 30, 1, 0, 14400, 21076, 7645, 1023, 55, 1, 0, 518400, 773136, 296296, 44473, 3003, 91, 1, 0, 25401600, 38402064, 15291640, 2475473, 191620, 7462, 140, 1

Offset: 0

Peter Luschny, Mar 22 2016

Also known as central factorial numbers |t(2*n, 2*k)| (cf. A008955).

The analog for the Stirling set numbers is A269945.

			Triangle starts:
  [1]
  [0,     1]
  [0,     1,     1]
  [0,     4,     5,    1]
  [0,    36,    49,   14,    1]
  [0,   576,   820,  273,   30,  1]
  [0, 14400, 21076, 7645, 1023, 55, 1]

Peter Luschny, The P-transform.
Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.
B. K. Miceli, Two q-Analogues of Poly-Stirling Numbers, J. Integer Seq., 14 (2011), 11.9.6.

Variants: A204579 (signed, row 0 missing), A008955.
Cf. A007318 (order 0), A132393 (order 1), A269947 (order 3).
Cf. A000330 (subdiagonal), A001044 (column 1), A101686 (row sums), A269945 (Stirling set), A269941 (P-transform).

T := proc(n, k) option remember; if n=k then return 1 fi; if k<0 or k>n then return 0 fi; T(n-1, k-1)+(n-1)^2*T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A269944_row := n -> PTrans(n, n->`if`(n=1, 1, (n-1)^2/(n*(4*n-2))), (n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269944_row(n)), n=0..8);
# From Peter Luschny, Feb 29 2024: (Start)
# Computed as the coefficients of polynomials:
P := (x, n) -> local j; mul((j - x)*(j + x), j = 0..n-1):
T := (n, k) -> (-1)^k*coeff(P(x, n), x, 2*k):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
# Alternative, using the exponential generating function:
egf := cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)):
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, n-k), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # (End)
# Alternative, row polynomials:
rowpoly := n -> pochhammer(-sqrt(x), n) * pochhammer(sqrt(x), n):
row := n -> local k; seq((-1)^k*coeff(expand(rowpoly(n)), x, k), k = 0..n):
seq(print(row(n)), n = 0..6);  # Peter Luschny, Aug 03 2024

T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + (n - 1)^2*T[n - 1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Jul 25 2019 *)

stircycle2 = lambda n: 1 if n == 1 else (n-1)^2/(n*(4*n-2))
norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stircycle2, norm)
for m in M: print(m)

T(n,k) = (-1)^k*((2*n)! / (2*k)!)*P[n, k](s(n)) where P is the P-transform and s(n) = (n - 1)^2 / (n*(4*n - 2)). The P-transform is defined in the link. See the Sage and Maple implementations below.
T(n, 1) = ((n - 1)!)^2 for n >= 1 (cf. A001044).
T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
Row sums: Product_{k=1..n} ((k - 1)^2 + 1) for n >= 0 (cf. A101686).
From Fabián Pereyra, Apr 25 2022: (Start)
T(n,k) = (-1)^(n-k)*Sum_{j=2*k..2*n} Stirling1(2*n,j)*binomial(j,2*k)*(n-1)^(j-2*k).
T(n,k) = Sum_{j=0..2*k} (-1)^(j - k)*Stirling1(n, j)*Stirling1(n, 2*k - j). (End)
From Peter Luschny, Feb 29 2024: (Start)
T(n, k) = (-1)^k*[x^(2*k)] P(x, n) where P(x, n) = Product_{j=0..n-1} (j-x)*(j+x).
T(n, k) = (2*n)!*[t^(n-k)] [x^(2*n)] cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)). (End)
T(n, k) = (-1)^k*[x^k] Pochhammer(-sqrt(x), n) * Pochhammer(sqrt(x), n). - Peter Luschny, Aug 03 2024

A048617 a(n) = 2*(n!)^2.

Original entry on oeis.org

2, 2, 8, 72, 1152, 28800, 1036800, 50803200, 3251404800, 263363788800, 26336378880000, 3186701844480000, 458885065605120000, 77551576087265280000, 15200108913103994880000, 3420024505448398848000000, 875526273394790105088000000, 253027093011094340370432000000

Offset: 0

N. J. A. Sloane

nonn

a(n) = automorphism group order for the complete bipartite graph K_{n,n}. - Avi Peretz (njk(AT)netvision.net.il), Feb 21 2001
For n > 1, also the order of automorphism group for the n X n rook graph. - Eric W. Weisstein, Jun 20 2017
Also the number of (directed) Hamiltonian paths in K_{n,n}. - Eric W. Weisstein, Jul 15 2011
For n>=1, a(n) is the number of ways to arrange n men and n women in a line so that no two people of the same gender are adjacent. - Geoffrey Critzer, Aug 24 2013
Also the number of (directed) Hamiltonian paths in the (n+1)-barbell graph. - Eric W. Weisstein, Dec 16 2013

Eric Weisstein's World of Mathematics, Graph Automorphism
Eric Weisstein's World of Mathematics, Barbell Graph
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Rook Graph

Equals 2 * A001044.

[2*Factorial(n)^2: n in [0..30]]; // Vincenzo Librandi, Feb 22 2016

seq(mul(n!*k!, k=1..2), n=0..17); # Zerinvary Lajos, Jul 01 2007

2(Range[0,20]!)^2 (* Harvey P. Dale, Jun 21 2011 *)
Table[2 (n!)^2, {n, 0, 20}] (* Vincenzo Librandi, Feb 22 2016 *)

a(n) = 2*A001044(n)

A239841 Ordered pairs of permutation functions on n elements where f(g(g(x))) = g(g(f(x))).

Original entry on oeis.org

1, 1, 4, 30, 312, 3720, 64080, 1305360, 33949440, 1019692800, 36360576000, 1487539468800, 69633899596800, 3649476307276800, 213929162589542400, 13848506938506240000, 986705192227442688000, 76724136092268048384000, 6491471142159880740864000

Offset: 0

Chad Brewbaker, Mar 27 2014

nonn

From Paul Boddington, Feb 24 2015: (Start)
Suppose G is the symmetric group on n letters. For each g in G, the set of f satisfying fgg = ggf is just the centralizer Z_gg(G). However |Z_gg(G)| is clearly constant on conjugacy classes of G. By the orbit-stabilizer theorem the size of the conjugacy class containing g is |G| / |Z_g(G)|. Since |G| = n! and Z_g(G) is a subgroup of Z_gg(G) we see that a(n) equals n! multiplied by the sum of indices |Z_gg(G) : Z_g(G)| where the sum is over representatives of the conjugacy classes of G. Since the conjugacy classes of G correspond to partitions of n (A000041), this makes it relatively easy to find terms.
a(n) appears to equal n! * A082733(n).
(End)

John F. Humphreys, A Course In Group Theory, Oxford Science Publications, 1996, chapter 10.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..60
Eric Weisstein's World of Mathematics, Symmetric Group

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

a(8)-a(9) from Giovanni Resta, Mar 27 2014

More terms from Paul Boddington, Feb 23 2015

A056039 Largest k such that (k!)^2 divides n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 15, 15, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 28, 28, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37

Offset: 1

Labos Elemer, Jul 25 2000

nonn

Let d(n) = a(n) - floor(n/2). Then, d(n) >= 0, and below 1000, d=1 arises 93 times, and d=2 arises 4 times.

The first occurrence of d(k) = 0, 1, 2, ..., is at k = 2, 1 (=A056067(1)), 416 (=A056068(1)), 6950, 16348, 505930, ... . - Amiram Eldar, May 24 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000142, A001057, A001405, A055772, A056038, A056067, A056068.

Cf. A001044, A105350.

a[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; k - 1]; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

A105350(n) = A001044(a(n)).

A089041 Inverse binomial transform of squares of factorial numbers.

Original entry on oeis.org

1, 0, 3, 26, 453, 11844, 439975, 22056222, 1436236809, 117923229512, 11921584264011, 1455483251191650, 211163237294447053, 35913642489947449356, 7077505637217289437423, 1599980633296779087784934, 411293643476907595937924625, 119299057697083019137937718672

Offset: 0

Vladeta Jovovic, Dec 03 2003

a(n) enumerates (ordered) lists of n two-tuples such that all numbers from 1 to n appear as the first as well as the second tuple entry and the j-th list member is not the tuple (j,j), for every j=1,..,n. Called coincidence-free 2-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.

			2-tuple combinatorics: a(1)=0 because the only list of 2-tuples with numbers 1 is [(1,1)] and this is a coincidence for j=1.
2-tuple combinatorics: the a(2)=3 coincidence free 2-tuple lists of length n=2 are [(1,2),(2,1)], [(2,1),(1,2)] and [(2,2),(1,1)]. The list [(1,1),(2,2)] has two coincidences (j=1 and j=2).

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=2.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013

Cf. A001044, A046662 (binomial transform of squares of factorial numbers).

(-1)^n times the polynomials in A099599 evaluated at -1.

a:= proc(n) option remember;
       `if`(n<2, 1-n, n^2*a(n-1)+n*(n-1)*a(n-2)+(-1)^n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 21 2013

Table[n!Sum[(-1)^k(n-k)!/k!,{k,0,n}],{n,0,15}] (* Geoffrey Critzer, Jun 17 2013 *)

G.f.: hypergeom([1, 1, 1], [], x/(1+x))/(1+x).
E.g.f.: exp(-x)* hypergeom([1, 1], [], x).
a(n) = n^2*a(n-1) + n*(n-1)*a(n-2) + (-1)^n. - Vladeta Jovovic, Jul 15 2004
a(n) = Sum_{j=0..n} ((-1)^(n-j))*binomial(n,j)*(j!)^2. See the Charalambides reference a(n)=B_{n,2}.
a(n) = (n-1)*(n+1)*a(n-1) + (n-1)*(2*n-1)*a(n-2) + (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 2*Pi*exp(-2*n)*n^(2*n+1). - Vaclav Kotesovec, Aug 13 2013
G.f.: Sum_{k>=0} (k!)^2*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019

Charalambides reference and comments with combinatorial examples from Wolfdieter Lang, Jan 21 2008

A134370 a(n) = ((2n+1)!)^(n+2).

Original entry on oeis.org

1, 216, 207360000, 3252016064102400000, 2283380023591730815784976384000000, 161469323688736156802100136913438716723200000000000000, 2260697901194635682690248130915498742378267539496871953338204160000000000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A134366, A134367, A134368, A134369, A134371, A134374, A134375.

Mathematica
```
Table[((2n+1)!)^(n + 2), {n, 0, 10}]
```

a(n) ~ 2^(2*(n+1)*(n+2)) * exp(13/24 - 2*n*(n+2)) * n^((n+2)*(4*n+3)/2) * Pi^(n/2 + 1). - Vaclav Kotesovec, Oct 26 2017

Typo in a(6) corrected by Georg Fischer, Apr 10 2024

cp's OEIS Frontend

A134434 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k even entries that are followed by a smaller entry (n>=0, k>=0).

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A166602 Numbers k such that Sum_{i=1..k} i^2 divides Product_{i=1..k} i^2.

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Maple

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A055546 a(n) = (-1)^(n+1) * 2^n * n!^2.

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PARI

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A134372 a(n) = ((2n)!)^2.

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PARI

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A269944 Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.

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Formula

A048617 a(n) = 2*(n!)^2.

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Magma

Maple

Mathematica

Formula

A239841 Ordered pairs of permutation functions on n elements where f(g(g(x))) = g(g(f(x))).

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