A002674 -id:A002674 - cp's OEIS frontend

A002676 Denominators of coefficients for central differences M_{4}^(2*n).

1, 6, 80, 30240, 1814400, 2661120, 871782912000, 3138418483200, 84687482880000, 170303140572364800, 1124000727777607680000, 724146127139635200000, 12703681025488077520896000000, 76222086152928465125376000000, 1531041037877004667453440000000

Offset: 2

N. J. A. Sloane

From Peter Bala, Oct 03 2019: (Start)
Denominators in the expansion of (2*sinh(x/2))^4 = x^4 + (1/6)*x^6 + (1/80)*x^8 + (17/30240)*x^10 + ....
Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^4 leads to a formula for the fourth central differences: f(x+2) - 4*f(x+1) + 6*f(x) - 4*f(x-1) + f(x-2) = (2*sinh(D/2))^4(f(x)) = D^4(f(x)) + (1/6)*D^6(f(x)) + (1/80)*D^8(f(x)) + (17/30240)*D^10(f(x)) + ..., where D denotes the differential operator d/dx. (End)

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).

H. E. Salzer, Tables of coefficients for obtaining central differences from the derivatives, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables.
H. E. Salzer, Annotated scanned copy of left side of Table II.
E. W. Weisstein, Central Difference. From MathWorld--A Wolfram Web Resource.

Cf. A002675 (numerators). Cf. A002671, A002672, A002673, A002674, A002677.

gf := 6 - 8*cosh(sqrt(x)) + 2*cosh(2*sqrt(x)): ser := series(gf, x, 40):
seq(denom(coeff(ser,x,n)), n=2..16); # Peter Luschny, Oct 05 2019

a(n) = denominator(4! * m(4, 2 * n) / (2 * n)!) where m(k, q) is defined in A002672. - Sean A. Irvine, Dec 20 2016

More terms from Sean A. Irvine, Dec 20 2016

A091543 Triangle built from first column sequences of generalized Stirling2 arrays (m+2,2)-Stirling2, m >= 0.

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 72, 12, 1, 16, 1440, 360, 20, 1, 32, 43200, 20160, 1120, 30, 1, 64, 1814400, 1814400, 123200, 2700, 42, 1, 128, 101606400, 239500800, 22422400, 491400, 5544, 56, 1, 256, 7315660800, 43589145600, 6098892800, 150368400

Offset: 1

Wolfdieter Lang, Feb 13 2004

P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Wolfdieter Lang, First 10 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A091547 (row sums), A091548 (alternating row sums).

For m = 0, 1, ..., 6 the column sequences are (without leading zeros): A000079 (powers of 2), A010796, A002674, A091535, A091544-6.

a(n, m) = m^(2*(n-m))*Pochhammer(1/m, n-m)*Pochhammer(2/m, n-m)/2 if n-1 >= m >= 1; a(n, 0) = 2^(n-1); otherwise 0.
E.g.f. for m = 1, 2, ... column (without leading zeros and offset n=1): (hypergeom([1/m, 2/m], [], (m^2)*x)-1)/2.
G.f. for m=1 column: x/(1-2*x); e.g.f.: (exp(2*x)-1)/2.
a(n, m) = (1/2)*Product_{j=0..n-m-1} (m*j+2)*(m*j+1), n >= m+1 >= 1, otherwise 0. From eq. 12 of the Blasiak et al. reference with r=m+2, s=2, k=2.

A211374 Product of all the parts in the partitions of n into exactly 2 parts.

Original entry on oeis.org

1, 1, 2, 12, 24, 360, 720, 20160, 40320, 1814400, 3628800, 239500800, 479001600, 43589145600, 87178291200, 10461394944000, 20922789888000, 3201186852864000, 6402373705728000, 1216451004088320000, 2432902008176640000, 562000363888803840000

Offset: 1

Wesley Ivan Hurt, Feb 06 2013

			Define a(1):=1; a(2) = 1 since 2 = 1+1 and (1)*(1) = 1; a(3) = 2 since 3 = 2+1 and (2)*(1) = 2; a(4) = 12 since 4 = 3+1 = 2+2 and (3)*(1)*(2)*(2) = 12; a(5) = 24 since 5 = 4+1 = 3+2 and (4)*(1)*(3)*(2) = 24.

G. C. Greubel, Table of n, a(n) for n = 1..449
Index entries for sequences related to partitions.

Cf. A001318, A002620, A081123, A081125, A093353.

Bisections: A002674, A010050.

[(Factorial(n-1) * Factorial(Floor(n/2)))/Factorial(n-1-Floor(n/2)) : n in [1..25]]; // Wesley Ivan Hurt, Oct 16 2014

A211374:=n->( (n-1)! * floor(n/2)! )/( (n-1) - floor(n/2) )!: seq(A211374(k), k=1..25);
with(combinat, numbperm): seq(numbperm(k-1, floor(k/2))*floor(k/2)!, k = 1..25); # Wesley Ivan Hurt, Jun 07 2013

Table[Times @@ Flatten[Select[Partitions[n], Length[#] == 2 &]], {n, 25}] (* T. D. Noe, Feb 11 2013 *)
Table[((n - 1)!*Floor[n/2]!)/(n - 1 - Floor[n/2])!, {n, 25}] (* Wesley Ivan Hurt, Oct 16 2014 *)

a(n) = prod(i=1, n\2, i*(n-i)); \\ Michel Marcus, Nov 14 2017

a(n) = ( (n-1)! * floor(n/2)! )/( n-1-floor(n/2) )!.
a(n) = P(n-1, floor(n/2)) * floor(n/2)!, where P(n,k) are the k-permutations of n objects. - Wesley Ivan Hurt, Jun 07 2013
a(2n) = A002674(n); a(2n+1) = A010050(n). - Wesley Ivan Hurt, Oct 16 2014
a(n) = Product_{i=1..floor(n/2)} i * (n-i). - Wesley Ivan Hurt, Nov 14 2017
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*cosh(1) - 2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - cosh(1). (End)

A186432 Triangle associated with the set S of squares {0,1,4,9,16,...}.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 30, 30, 1, 1, 56, 140, 56, 1, 1, 90, 420, 420, 90, 1, 1, 132, 990, 1848, 990, 132, 1, 1, 182, 2002, 6006, 6006, 2002, 182, 1, 1, 240, 3640, 16016, 25740, 16016, 3640, 240, 1, 1, 306, 6120, 37128, 87516, 87516, 37128, 6120, 306, 1, 1, 380, 9690, 77520, 251940, 369512, 251940, 77520, 9690, 380, 1

Offset: 0

Peter Bala, Feb 22 2011

Given a subset S of the integers Z, Bhargava [1] has shown how to associate with S a generalized factorial function, denoted n!_S, sharing many properties of the classical factorial function n! (which corresponds to the choice S = Z). In particular, he shows that the generalized binomial coefficients n!_S/(k!_S*(n-k)!_S) are always integral for any choice of S. Here we take S = {0,1,4,9,16,...}, the set of squares.
The associated generalized factorial function n!_S is given by the formula
n!S = Product{k=0..n} (n^2 - k^2), with the convention 0!S = 1. This should be compared with n! = Product{k=0..n} (n - k).
For n >= 1, n!_S = (2*n)!/2 = A002674(n).
Compare this triangle with A086645 and also A186430 - the generalized binomial coefficients for the set S of prime numbers {2,3,5,7,11,...}.

			Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1....12.....1
.3..|..1....30....30.....1
.4..|..1....56...140....56.....1
.5..|..1....90...420...420....90.....1
.6..|..1...132...990..1848...990...132.....1
.7..|..1...182..2002..6006..6006..2002...182.....1
...

M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (2000), 783-799.

Cf. A002114, A086645, A186430, A186433 (inverse).

Table[2 Binomial[2 n, 2 k] - Boole[Or[k == 0, k == n]], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 23 2017 *)

TABLE ENTRIES
T(n,k) = n!_S/(k!_S*(n-k)!_S),
which simplifies to
T(n,k) = 2*binomial(2*n,2*k) for 1 <= k < n,
with boundary conditions T(n,0) = 1 and T(n,n) = 1 for n >= 0.
RELATIONS WITH OTHER SEQUENCES
Denote this triangle by T. The first column of the inverse T^-1 (see A186433) begins [1, -1, 11, -301, 15371, ...] and, apart from the initial 1, is a signed version of the Glaisher's H' numbers A002114.
The first column of (1/2)*T^2 begins [1/2, 1, 7, 31, 127, ...] and, apart from the initial term, equals A000225(2*n-1), counting the preferential arrangements on (2*n - 1) labeled elements having less than or equal to two ranks.
The first column of (1/3)*T^3 begins [1/3, 1, 13, 181, 1933, ...] and, apart from the initial term, is A101052(2*n-1), which gives the number of preferential arrangements on (2*n-1) labeled elements having less than or equal to three ranks.

A191662 a(n) = n! / A000034(n-1).

Original entry on oeis.org

1, 1, 6, 12, 120, 360, 5040, 20160, 362880, 1814400, 39916800, 239500800, 6227020800, 43589145600, 1307674368000, 10461394944000, 355687428096000, 3201186852864000, 121645100408832000, 1216451004088320000, 51090942171709440000, 562000363888803840000

Offset: 1

Paul Curtz, Jun 10 2011

The a(n) are the denominators in the formulas of the k-dimensional square pyramidal numbers:
A005408 = (2*n+1)/1                       = 1, 3,  5,   7,   9, ... (k=1)
A000290 = (n^2)/1                         = 1, 4,  9,  16,  25, ... (k=2)
A000330 = n*(n+1)*(2*n+1)/6               = 1, 5, 14,  30,  55, ... (k=3)
A002415 = (n^2)*(n^2-1)/12                = 1, 6, 20,  50, 105, ... (k=4)
A005585 = n*(n+1)*(n+2)*(n+3)*(2*n+3)/120 = 1, 7, 27,  77, 182, ... (k=5)
A040977 = (n^2)*(n^2-1)*(n^2-4)/360       = 1, 8, 35, 112, 294, ... (k=6)
A050486 (k=7), A053347 (k=8), A054333 (k=9), A054334 (k=10), A057788 (k=11).
The first superdiagonal of this array appears in A029651. - Paul Curtz, Jul 04 2011
The general formula for the k-dimensional square pyramidal numbers is (2*n+k)*binomial(n+k-1,k-1)/k, k >= 1, n >= 0, see A097207. - Johannes W. Meijer, Jun 22 2011

Cf. A000142, A009445, A002674, A064680.
Cf. A000290, A000330, A002415, A005408, A005585, A029651, A040977, A050486, A053347, A054333, A054334, A057788.
Cf. A052612.

A191662:= proc(n): n!/A000034(n-1) end: A000034 := proc(n) op((n mod 2)+1, [1, 2]) ; end proc: seq(A191662(n),n=1..17); # Johannes W. Meijer, Jun 22 2011

Array[If[EvenQ[#],#!/2,#!]&,20] (* Harvey P. Dale, Mar 14 2014 *)

a(2*n-1) = (2*n-1)!, a(2*n) = (2*n)!/2.
a(n+1) = A064680(n+1) * a(n).
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = sinh(1) + 2*cosh(1) - 2.
Sum_{n>=1} (-1)^(n+1)/a(n) = sinh(1) - 2*cosh(1) + 2. (End)
D-finite with recurrence: a(n) - (n-1)*n*a(n-2) = 0 for n >= 3 with a(1)=a(2)=1. - Georg Fischer, Nov 25 2022
a(n) = A052612(n)/2 for n >= 1. - Alois P. Heinz, Sep 05 2023

More terms from Harvey P. Dale, Mar 14 2014

A328826 Triangle read by rows: binomial(n,k)(2n-k)!, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 2, 1, 24, 12, 2, 720, 360, 72, 6, 40320, 20160, 4320, 480, 24, 3628800, 1814400, 403200, 50400, 3600, 120, 479001600, 239500800, 54432000, 7257600, 604800, 30240, 720, 87178291200, 43589145600, 10059033600, 1397088000, 127008000, 7620480, 282240, 5040, 20922789888000

Offset: 0

R. J. Mathar, Oct 28 2019

Vertex-labeled disconnected Goldstone diagrams with n vertices and k single-particle potentials.

			The triangle starts
    1;
    2     1;
   24    12     2;
  720   360    72     6;
40320 20160  4320   480    24;

P. J. Rossky, M. Karplus, The enumeration of Goldstone diagrams in many-body perturbation theory, J. Chem. Phys. 64 (1976) 1569, equation (9) and Table 1.

Cf. A099022 (row sums), A000142 (diagonal), A010050 (column k=0), A002674 (k=1).

A328826 := proc(n,k)
        binomial(n,k)*(2*n-k)! ;
end proc:

Table[Binomial[n,k](2n-k)!,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Feb 03 2022 *)

T(n,k)= binomial(n,k)*(2*n-k)!.

T(n,k) = A328921(n,k) + A328922(n,k). - R. J. Mathar, Nov 02 2019

A370707 Triangle read by rows: T(n, k) = (-1)^kProduct_{j=0..k-1} (j - n)(j + n), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 12, 1, 9, 72, 360, 1, 16, 240, 2880, 20160, 1, 25, 600, 12600, 201600, 1814400, 1, 36, 1260, 40320, 1088640, 21772800, 239500800, 1, 49, 2352, 105840, 4233600, 139708800, 3353011200, 43589145600, 1, 64, 4032, 241920, 13305600, 638668800, 24908083200, 697426329600, 10461394944000

Offset: 0

Peter Luschny, Feb 27 2024

The definition, and also the representation T(n, k) = ff(n, k) * rf(n, k) (see the first formula), makes it natural to call this triangle the central factorial numbers.

			Triangle starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  4,   12;
  [3] 1,  9,   72,    360;
  [4] 1, 16,  240,   2880,   20160;
  [5] 1, 25,  600,  12600,  201600,   1814400;
  [6] 1, 36, 1260,  40320, 1088640,  21772800,  239500800;
  [7] 1, 49, 2352, 105840, 4233600, 139708800, 3353011200, 43589145600;
.
T(n, k) is a product where 'n' is the 'center' and 'k' is the 'half-length' of the product. For instance, T(5, 4) = (5-3)*(5-2)*(5-1)*5 * 5*(5+1)*(5+2)*(5+3) = 201600. Now consider the polynomial P(4, x) = -36*x^2 + 49*x^4 - 14*x^6 + x^8. Evaluating this polynomial at x = 5 shows P(4, 5) = 201600 = T(5, 4). The coefficients of the polynomial are row 4 of A269944.

Diagonals: A002674, A327882.
Columns: A000290, A047928.
Cf. A370704 (row sums), A370706, A094728, A048994 (Stirling1), A130595 (order 0), A269947 (order 3)

T := (n, k) -> local j; (-1)^k * mul((j - n)*(j + n), j = 0..k-1):
seq(seq(T(n, k), k = 0..n), n = 0..8);
# The central factorial numbers:
cf := (n, k) -> ifelse(k = 0, 1, n*(n + k - 1)! / (n - k)! ):
for n from 0 to 6 do seq(cf(n, k), k = 0..n) od;
# Alternative (recurrence):
T := proc(n, k) option remember;
if k = 0 then 1 else T(n, k - 1)*(n^2 - (k - 1)^2) fi end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;
# Illustrating the connection with the cf-polynomials and their coefficients:
cfpoly := (n,x) -> local k; mul(x^2 - k^2, k = 0..n-1):
A370707row := n -> local k; [seq(cfpoly(k, n), k = 0..n)]:
A204579row := n -> local k; [seq(coeff(cfpoly(n, x), x, 2*k), k = 0..n)]:
for n from 0 to 5 do lprint([n], A370707row(n), A204579row(n)) od;

T[n_, k_] := If[n == 0, 1, -n Pochhammer[1 - n - k, 2 k - 1]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

from math import prod
def T(n, k): return (-1)**k * prod((j - n)*(j + n) for j in range(k))
print([T(n, k) for n in range(8) for k in range(n + 1)])

def T(n, k): return falling_factorial(n, k) * rising_factorial(n, k)
for n in range(9): print([T(n, k) for k in range(n + 1)])

T(n, k) = FallingFactorial(n, k) * RisingFactorial(n, k).
T(n, k) = (n*(n + k - 1)!)/(n - k)! if k > 0, and T(n, 0) = 1.
Calling the numbers in the second formula cf leads to the memorable form cf(n, k) = ff(n, k) * rf(n, k). This identity generalizes to the function
  cf(x, n) = x*Gamma(x + n)/Gamma(x - n + 1) for n > 0 and cf(x, 0) = 1.
The last equation shows that the variable 'n' does not have to be an integer but can be any complex number if only the quotient remains defined (which one often can achieve by taking the limit). Indeed, in the classical Steffensen-Riordan case, n/2 is used instead of n, which leads to the complex situation Sloane discusses in A008955.
T(n, k) = -n*Pochhammer(1 - n - k, 2*k - 1) for n > 0.
T(n, k) = k!*binomial(n, k)*Pochhammer(n, k) = k!*A370706(n, k).
T(n, n) = n!*Pochhammer(n, n) (valid for n >= 0, whereas T(n, n) = (2*n)!/2 = A002674(n) is valid for n >= 1 only).
T(n, k) = T(n, k - 1)*(n^2 - (k - 1)^2) if k > 0, otherwise 1. (Recurrence)
The cf(n, k) are values of the polynomials Pcf(n, x) = Product_{k=0..n-1} (x^2 - k^2), whose coefficients vanish for odd powers and for even powers are A269944.
T(n, k) = Pcf(k, n) where Pcf(k,x) = Sum_{j=0..k} (-1)^(k-j)*A269944(k,j)*x^(2*j).
The central factorials can be described in three different ways: By the product T(n, k) = f(n, k) * rf(n, k), by the complex function cf(x, n), and through the polynomials Pcf(n, x). Although these relations are self-contained, they are regarded as only one-half of a more general notion, namely as central factorials of the first kind.
There is a fundamental connection with the Stirling numbers of first kind (A048994). The easiest way to see this is to generalize the definition: Let CF(z, s) = Product_{j=0..n-1} (z - s(j)), where s(j) is some complex sequence. Then the coefficients of CF(z, s) are equal to the Stirling_1 numbers if s = 0, 1, 2, ..., n, ..., and they are equal to the coefficients of our Pcf(n, z) polynomials if s = 0, 1, 4, ..., n^2, .... (This is also why A269944 is called the 'Stirling cycle numbers of order 2'. For completeness, if s = 1, 1, 1, ..., then the coefficients of CF(z, s), the 'Stirling cycle numbers of order 0', are the signed Pascal triangle A130595. See A269947 for order 3.)

A091032 Second column (k=3) of array A090438 ((4,2)-Stirling2) divided by 8.

Original entry on oeis.org

1, 60, 5040, 604800, 99792000, 21794572800, 6102480384000, 2134124568576000, 912338253066240000, 468333636574003200000, 284372184127734743040000, 201645730563302817792000000, 165147853331345007771648000000

Offset: 2

Wolfdieter Lang, Jan 23 2004

John Campbell, Applications of Gosper’s nonlocal derangement identity, Maple Transactions, 5, 1, Article 16724, Feb. 2025.

Cf. A002674 (first column of A090438), A091033 (third column), A090438.

Cf. A001620, A049470, A073742, A073743, A099281, A099282, A099284.

a[n_] := (n - 1)*(2*n)!/4!; Array[a, 13, 2] (* Amiram Eldar, Nov 03 2022 *)

a(n) = (n-1)*(2*n)!/4!; \\ Amiram Eldar, Nov 03 2022

a(n) = A090438(n, 3)/8 = (n-1)*(2*n)!/4!
E.g.f.: (-3*hypergeom([1/2, 1], [], 4*x) + hypergeom([1, 3/2], [], 4*x) + 2)/(8*3!) (cf. A090438).
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=2} 1/a(n) = 60 - 24*Gamma - 24*cosh(1) + 24*CoshIntegral(1) - 24*sinh(1).
Sum_{n>=2} (-1)^n/a(n) = -12 + 24*gamma - 24*cos(1) - 24*CosIntegral(1) + 24*SinIntegral(1). (End)
a(n+1) = Sum_{j = 1..n} (-1)^(n+j) * j^(2*n+2) * binomial(2*n, n-j) (Campbell, Eq. 17). - Peter Bala, Mar 30 2025

A128103 Number of permutations of [n] with an even number of rises.

Original entry on oeis.org

1, 1, 1, 2, 12, 68, 360, 2384, 20160, 185408, 1814400, 19781504, 239500800, 3124694528, 43589145600, 652885305344, 10461394944000, 177948646719488, 3201186852864000, 60808005761859584, 1216451004088320000, 25547946834881282048, 562000363888803840000

Offset: 0

Ralf Stephan, May 09 2007

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400
F. C. S. Brown, T. M. A. Fink and K. Willbrand, On arithmetic and asymptotic properties of up-down numbers, arXiv:math/0607763 [math.CO], 2006.

Cf. A000142, A002674, A002430, A036279, A262745.

b:= proc(u, o, t) option remember; `if`(u+o=0, t,
      add(b(u-j, o+j-1, t), j=1..u)+
      add(b(u+j-1, o-j, 1-t), j=1..o))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 29 2015

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, t, Sum[b[u - j, o + j - 1, t], {j, 1, u}] + Sum[b[u + j - 1, o - j, 1 - t], {j, 1, o}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 25 2017, after Alois P. Heinz *)

x='x+O('x^99); Vec(serlaplace((x/(1-x)+tanh(x))/2+1)) \\ Altug Alkan, Jul 25 2017

E.g.f.: 1 + 1/2 [z/(1-z) + tanh(z) ].
a(n) = A000142(n) - A262745(n).
If n is even, a(n) = (n)!/2 (A002674), if n is odd, a(n) = (n)! * (1 + (-1)^((n-1)/2) * A002430((n+1)/2) / A036279((n+1)/2)) / 2. - Michel Marcus, Dec 09 2012
Conjecture: a(n) = Sum_{k = 0..n} Sum_{j = 0..k} (-1)^(n+j)*binomial(n,k-j)*j^n. - Peter Bala, Jan 22 2020

More terms from Alois P. Heinz, Sep 29 2015

A159960 a(n) = (1/2) * Sum_{k=1..n} (-1)^(k-1) * binomial(2n-k, k) binomial(n, k) * 2^k * (2n-2k)!.

Original entry on oeis.org

1, 10, 292, 16152, 1443616, 189709600, 34420171584, 8241995095936, 2517637537094656, 955377719901439488, 440888939541736115200, 243144648530111594371072, 157920570527279020394569728, 119308432982412667510831095808, 103738687936577909824307104989184

Offset: 1

Ji Li (vieplivee(AT)hotmail.com), Apr 28 2009

Previous name was "Number of permutations of the set 1,2,..., 2n such that at least one pair of adjacent numbers in the permutation differs by n.", which did not match data. See A386965. - Giovanni Resta, Aug 12 2025

G. C. Greubel, Table of n, a(n) for n = 1..220

Cf. A002674, A386965.

f := proc (n) add((-1)^(k-1)*binomial(2*n-k, k)*binomial(n, k)*2^k*factorial(2*n-2*k), k = 1 .. n)/2 end proc;

a[n_] := (2*n)!*(1-HypergeometricPFQ[{-n}, {1, -2*n}, -2])/2; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Jan 27 2014 *)

a(n)=sum(k=1,n,(-1)^(k-1)*binomial(2*n-k,k)*binomial(n, k)<Charles R Greathouse IV, Jun 19 2013

a(n) = (1/2) * Sum_{k=1..n} (-1)^(k-1) * binomial(2*n-k, k) * binomial(n, k) * 2^k * (2*n-2*k)!.
Recurrence: (6*n - 17)*a(n) = 2*(n-1)*(36*n^2 - 156*n + 151)*a(n-1) - 4*(n-1)*(72*n^4 - 636*n^3 + 2062*n^2 - 2909*n + 1511)*a(n-2) + 4*(n-2)*(n-1)*(96*n^5 - 1280*n^4 + 6704*n^3 - 17208*n^2 + 21596*n - 10569)*a(n-3) + 8*(n-3)*(n-2)*(n-1)*(2*n - 7)*(6*n - 11)*a(n-4). - Vaclav Kotesovec, Mar 15 2014
a(n) ~ (1-BesselJ(0,2)) * sqrt(Pi) * 4^n * n^(2*n+1/2) / exp(2*n). - Vaclav Kotesovec, Mar 15 2014

Name corrected and edited by Giovanni Resta, Aug 12 2025

cp's OEIS Frontend

A002676 Denominators of coefficients for central differences M_{4}^(2*n).

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A091543 Triangle built from first column sequences of generalized Stirling2 arrays (m+2,2)-Stirling2, m >= 0.

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A211374 Product of all the parts in the partitions of n into exactly 2 parts.

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A186432 Triangle associated with the set S of squares {0,1,4,9,16,...}.

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A191662 a(n) = n! / A000034(n-1).

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A328826 Triangle read by rows: binomial(n,k)*(2*n-k)!, n>=0, 0<=k<=n.

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A370707 Triangle read by rows: T(n, k) = (-1)^k*Product_{j=0..k-1} (j - n)*(j + n), for 0 <= k <= n.

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A328826 Triangle read by rows: binomial(n,k)(2n-k)!, n>=0, 0<=k<=n.

A370707 Triangle read by rows: T(n, k) = (-1)^kProduct_{j=0..k-1} (j - n)(j + n), for 0 <= k <= n.

A159960 a(n) = (1/2) * Sum_{k=1..n} (-1)^(k-1) * binomial(2n-k, k) binomial(n, k) * 2^k * (2n-2k)!.