A010790 -id:A010790 - cp's OEIS frontend

A047677 Row 2 of square array defined in A047675: 2n!(n+1)!.

2, 4, 24, 288, 5760, 172800, 7257600, 406425600, 29262643200, 2633637888000, 289700167680000, 38240422133760000, 5965505852866560000, 1085722065221713920000, 228001633696559923200000, 54720392087174381568000000, 14883946647711431786496000000

Offset: 0

N. J. A. Sloane

nonn

a(n) = A152877(2n+1, 2n-2) for n > 0. - Alois P. Heinz, Nov 10 2013

Cf. A047675, A010790, A152877.

a:= proc(n) a(n):= `if`(n=0, 2, n*(n+1) * a(n-1)) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 11 2013

2*Times@@@Partition[Range[0,20]!,2,1] (* Harvey P. Dale, Sep 25 2017 *)

A129274 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk+k) in the squared q-factorial of n+1.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 71, 71, 1, 1, 474, 1930, 474, 1, 1, 3103, 40096, 40096, 3103, 1, 1, 20190, 739929, 2108560, 739929, 20190, 1, 1, 131204, 12836959, 88638236, 88638236, 12836959, 131204, 1, 1, 853176, 215022825, 3286786158, 7625997280

Offset: 0

Paul D. Hanna, Apr 07 2007

Row sums equal A010790(n) = n!*(n+1)! for n>=0. Central terms form a bisection of A127728. Dual triangle is A129276.

			Definition of q-factorial of n:
faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
Obtain row 3 from coefficients in the squared q-factorial of 4:
faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
= (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
the resulting coefficients of q are:
[(1), 6, 19, 42, (71), 96, 106, 96, (71), 42, 19, 6, (1)],
where the terms enclosed in parenthesis form row 3.
Triangle begins:
1;
1, 1;
1, 10, 1;
1, 71, 71, 1;
1, 474, 1930, 474, 1;
1, 3103, 40096, 40096, 3103, 1;
1, 20190, 739929, 2108560, 739929, 20190, 1;
1, 131204, 12836959, 88638236, 88638236, 12836959, 131204, 1;
1, 853176, 215022825, 3286786158, 7625997280, 3286786158, 215022825, 853176, 1; ...

Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A129275 (column 1); A127728 (central terms), A010790 (row sums); related triangles: A129276, A128564, A008302 (Mahonian numbers).

T(n,k)=polcoeff(prod(i=1,n+1,(1-x^i)/(1-x))^2,(n+1)*k)

T(n,k) = [q^(nk+k)] Product_{i=1..n+1} { (1-q^i)/(1-q) }^2.

A172286 Numbers of circuits of length 2n in K_{n,n} (the complete bipartite graph on 2n vertices).

Original entry on oeis.org

2, 32, 1458, 131072, 19531250, 4353564672, 1356446145698, 562949953421312, 300189270593998242, 200000000000000000000, 162805498773679522226642, 158993694406781688266883072, 183466660386537233316799232018

Offset: 1

Thibaut Lienart (syncthib(AT)gmail.com), Jan 30 2010

Circuits are allowed to be self-intersecting and are directional with a designated start node. The number of (self-avoiding) directed cycles is given by A010790. - Andrew Howroyd, Sep 05 2018

			a(2) = 32 because there are 32 circuits of length 4 in the complete bipartite graph K2,2.

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A010790, A118537.

nmax = 10;
for k=1:nmax
an = 2*k^(2*k);
fprintf('%3.0f ', an);
end

a(n)=2*n^(2*n); \\ Andrew Howroyd, Sep 05 2018

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

More terms from Max Alekseyev, Jan 18 2012

A176037 a(n) = n!(n+1)!(n+2)!.

Original entry on oeis.org

2, 12, 288, 17280, 2073600, 435456000, 146313216000, 73741860864000, 53094139822080000, 52563198423859200000, 69383421919494144000000, 119061952013851951104000000, 260031303198252661211136000000, 709885457731229765106401280000000, 2385215137976932010757508300800000000

Offset: 0

Jonathan Vos Post, Apr 07 2010

			a(2) = (2)!*(2+1)!*(2+2)! = (2)!*(3)!*(4)! = 2*6*24 = 288.

Cf. A000142, A010790, A090443.

Times@@@Partition[Range[0,15]!,3,1] (* Harvey P. Dale, Aug 29 2012 *)

a(n) = A000142(n)*A000142(n+1)*A000142(n+2) = A010790(n)*A000142(n+2).

a(n) = 2*A090443(n). - Pontus von Brömssen, Jun 14 2024

A334174 Numbers that can be written as a product of two or more consecutive factorial numbers.

Original entry on oeis.org

1, 2, 12, 144, 288, 2880, 17280, 34560, 86400, 2073600, 3628800, 12441600, 24883200, 203212800, 435456000, 10450944000, 14631321600, 62705664000, 125411328000, 146313216000, 1316818944000, 17557585920000, 73741860864000, 144850083840000, 421382062080000

Offset: 1

Ilya Gutkovskiy, Apr 17 2020

nonn

			    1 = 0! * 1!;
    2 = 1! * 2!;
   12 = 2! * 3!;
  144 = 3! * 4!;
  288 = 2! * 3! * 4!.

David A. Corneth, Table of n, a(n) for n = 1..1803 (terms < 10^1000)
Eric Weisstein's World of Mathematics, Factorial Products
Index entries for sequences related to factorial numbers

Cf. A000142, A000178, A001013, A010790, A034882, A045619, A058295, A334175.

A368433 a(n) is the number of reduced instances in the stable marriage problem of order n that generate the maximum possible number of stable matchings.

Original entry on oeis.org

1, 1, 91, 1, 176130

Offset: 1

Dan Eilers, Dec 24 2023

Reduced instances (A351409) are fewer than all instances by a factor of n!(n-1)! due to participant-renaming isomorphism, analogous to reduced latin squares.

For n in [1,2,4], a(n) = 1 showing uniqueness up to isomorphism.

David F. Manlove, Algorithmics of Matching Under Preferences, World Scientific (2013) [Section 2.2.2].

Cf. A344669 (unreduced), A351430 (order 4), A368419 (order 5), A351409 (total reduced instances), A010790 (reduction factor offset by 1).

a(n) = A344669(n) / A010790(n-1).
a(4) = A351430(10).
a(5) = A368419(0).

A371767 Triangle read by rows: T(n, k) = (k! * n!)/(n - k)!.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 12, 36, 1, 4, 24, 144, 576, 1, 5, 40, 360, 2880, 14400, 1, 6, 60, 720, 8640, 86400, 518400, 1, 7, 84, 1260, 20160, 302400, 3628800, 25401600, 1, 8, 112, 2016, 40320, 806400, 14515200, 203212800, 1625702400

Offset: 0

Peter Luschny, Apr 14 2024

			Triangle starts:
  [0] 1;
  [1] 1, 1;
  [2] 1, 2,  4;
  [3] 1, 3, 12,   36;
  [4] 1, 4, 24,  144,   576;
  [5] 1, 5, 40,  360,  2880,  14400;
  [6] 1, 6, 60,  720,  8640,  86400,  518400;
  [7] 1, 7, 84, 1260, 20160, 302400, 3628800, 25401600;

Cf. A000142, A001044 (main diagonal), A010790 (subdiagonal), A046662 (row sums), A089041 (alternating row sums), A010050 (central terms).

T := (n, k) -> (k! * n!)/(n - k)!:
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;

A016065 a(n) = Sum_{k=0..n} k!*(k+1)!.

Original entry on oeis.org

1, 3, 15, 159, 3039, 89439, 3718239, 206931039, 14838252639, 1331657196639, 146181741036639, 19266392807916639, 3002019319241196639, 545863051930098156639, 114546679900210059756639, 27474742723487400843756639, 7469448066579203294091756639, 2284713285166428266627979756639

Offset: 0

Robert G. Wilson v

nonn

Partial sums of A010790. - Sean A. Irvine, Jan 02 2019

Seiichi Manyama, Table of n, a(n) for n = 0..252

Cf. A010790.

[&+[ Factorial(k)*Factorial(k+1): k in [0..n]]: n in [0..18]]; // Vincenzo Librandi, Jan 02 2019

Table[Sum[k! (k+1)!, {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Jan 02 2019 *)

a(n) = sum(k=0, n, k!*(k+1)!); \\ Michel Marcus, Jan 02 2019

More terms from Vincenzo Librandi, Jan 02 2019

A069135 a(n) = (n!*(n+1)!)^2.

Original entry on oeis.org

1, 4, 144, 20736, 8294400, 7464960000, 13168189440000, 41295442083840000, 214075571762626560000, 1734012131277275136000000, 20981546788455029145600000000, 365582471242040427832934400000000, 8896815020146295851742291558400000000, 294698100727325903793111665580441600000000

Offset: 0

Rebecca Gladu (rgladu(AT)eve.assumption.edu), Apr 07 2002

nonn

Cf. A010790, A071951.

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

a(n) = (n!*(n+1)!)^2; \\ Michel Marcus, Jan 15 2023

a(n) = det(PS(i+2,j), 1 <= i,j <= n), where PS(n,k) are Legendre-Stirling numbers of the second kind (A071951) [Mircea Merca, Apr 04 2013]

a(n) = A010790(n)^2. - Michel Marcus, Jan 15 2023

Edited and extended by Robert G. Wilson v, Apr 08 2002

More terms from Michel Marcus, Jan 15 2023

A176038 Primes of the form k!(k+1)! - 1 or k!(k+1)! + 1.

Original entry on oeis.org

2, 3, 11, 13, 2879, 86399, 114000816848279961600001, 2284848632399058501374484565150666260597460935294482959564800000000000001

Offset: 1

Jonathan Vos Post, Apr 07 2010

Primes of the form A010790(k)-1 or A010790(k)+1. This is the 2nd sequence in the supersequence whose first member is factorial primes, A002981 UNION A002982. No more through 20!*(20+1)! + 1.

a(9) has already 225 digits. The terms are generated by n= 0,1,2,2,4,5,14,32,76,166... [From R. J. Mathar, Aug 31 2010]

			a(6) = 86399 because 5!*(5+1)! - 1 = 86399 is prime. a(7) = 114000816848279961600001 because 14!*(14+1)! + 1 = 114000816848279961600001 is prime.

Cf. A000040, A000142, A002981, A002982, A010790.

[{A010790(n)-1} INTERSECTION A000040] UNION [{A010790(n)+1} INTERSECTION A000040].

One more term from R. J. Mathar, Aug 31 2010

cp's OEIS Frontend

A047677 Row 2 of square array defined in A047675: 2*n!*(n+1)!.

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A129274 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk+k) in the squared q-factorial of n+1.

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A172286 Numbers of circuits of length 2n in K_{n,n} (the complete bipartite graph on 2n vertices).

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A176037 a(n) = n!*(n+1)!*(n+2)!.

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A334174 Numbers that can be written as a product of two or more consecutive factorial numbers.

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A368433 a(n) is the number of reduced instances in the stable marriage problem of order n that generate the maximum possible number of stable matchings.

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A371767 Triangle read by rows: T(n, k) = (k! * n!)/(n - k)!.

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Maple

A016065 a(n) = Sum_{k=0..n} k!*(k+1)!.

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Magma

Mathematica

PARI

Extensions

A069135 a(n) = (n!*(n+1)!)^2.

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A047677 Row 2 of square array defined in A047675: 2n!(n+1)!.

A176037 a(n) = n!(n+1)!(n+2)!.

A176038 Primes of the form k!(k+1)! - 1 or k!(k+1)! + 1.