A080046 -id:A080046 - cp's OEIS frontend

A052849 a(0) = 0; a(n) = 2*n! (n >= 1).

0, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000, 2248001455555215360000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

For n >= 1 a(n) is the size of the centralizer of a transposition in the symmetric group S_(n+1). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001
For n > 0, a(n) = n! - A062119(n-1) = number of permutations of length n that have two specified elements adjacent. For example, a(4) = 12 as of the 24 permutations, 12 have say 1 and 2 adjacent: 1234, 2134, 1243, 2143, 3124, 3214, 4123, 4213, 3412, 3421, 4312, 4321. - Jon Perry, Jun 08 2003
With different offset, denominators of certain sums computed by Ramanujan.
From Michael Somos, Mar 04 2004: (Start)
Stirling transform of a(n) = [2, 4, 12, 48, 240, ...] is A000629(n) = [2, 6, 26, 150, 1082, ...].
Stirling transform of a(n-1) = [1, 2, 4, 12, 48, ...] is A007047(n-1) = [1, 3, 11, 51, 299, ...].
Stirling transform of a(n) = [1, 4, 12, 48, 240, ...] is A002050(n) = [1, 5, 25, 149, 1081, ...].
Stirling transform of 2*A006252(n) = [2, 2, 4, 8, 28, ...] is a(n) = [2, 4, 12, 48, 240, ...].
Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 2*A005649(n) = [4, 16, 88, 616, ...].
Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 4*A083410(n) = [4, 16, 88, 616, ...]. (End)
Number of {12, 12*, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
Permanent of the (0, 1)-matrices with (i, j)-th entry equal to 0 if and only if it is in the border but not the corners. The border of a matrix is defined the be the first and the last row, together with the first and the last column. The corners of a matrix are the entries (i = 1, j = 1), (i = 1, j = n), (i = n, j = 1) and (i = n, j = n). - Simone Severini, Oct 17 2004

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 520.

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 490.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 817.
Anna Khmelnitskaya, Gerard van der Laan, and Dolf Talmanm, The Number of Ways to Construct a Connected Graph: A Graph-Based Generalization of the Binomial Coefficients, J. Int. Seq. (2023) Art. 23.4.3. See p. 11.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for sequences related to factorial base representation.

Essentially the same sequence as A098558.
Cf. A062119, A080046, A245334, A000142.
Row 3 of A276955 (from term a(2)=4 onward).

a052849 n = if n == 0 then 0 else 2 * a000142 n
a052849_list = 0 : fs where fs = 2 : zipWith (*) [2..] fs
-- Reinhard Zumkeller, Aug 31 2014

[0] cat [2*Factorial(n-1): n in [2..25]]; // Vincenzo Librandi, Nov 03 2014

spec := [S,{B=Cycle(Z),C=Cycle(Z),S=Union(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0}, 2Range[20]!] (* Harvey P. Dale, Jul 13 2013 *)

PARI
```
a(n)=if(n<1,0,n!*2)
```

a(n) = T(n, 2) for n>1, where T is defined as in A080046.
D-finite with recurrence: {a(0) = 0, a(1) = 2, (-1 - n)*a(n+1) + a(n+2)=0}.
E.g.f.: 2*x/(1-x).
a(n) = A090802(n, n - 1) for n > 0. - Ross La Haye, Sep 26 2005
For n >= 1, a(n) = (n+3)!*Sum_{k=0..n+2} (-1)^k*binomial(2, k)/(n + 3 - k). - Milan Janjic, Dec 14 2008
G.f.: 2/Q(0) - 2, where Q(k) = 1 - x*(k + 1)/(1 - x*(k + 1)/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Apr 01 2013
G.f.: -2 + 2/Q(0), where Q(k) = 1 + k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
G.f.: W(0) - 2 , where W(k) = 1 + 1/( 1 - x*(k+1)/( x*(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013
a(n) = A245334(n, n-1), n > 0. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=1} 1/a(n) = (e-1)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (e-1)/(2*e). (End)

More terms from Ross La Haye, Sep 26 2005

A079542 Triangular array: T(n,1) = T(n,n) = n and T(n,k) = lcm(T(n-1,k-1), T(n-1,k)) for 1 < k < n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 12, 6, 12, 5, 6, 60, 12, 12, 60, 6, 7, 60, 60, 12, 60, 60, 7, 8, 420, 60, 60, 60, 60, 420, 8, 9, 840, 420, 60, 60, 60, 420, 840, 9, 10, 2520, 840, 420, 60, 60, 420, 840, 2520, 10, 11, 2520, 2520, 840, 420, 60, 420, 840, 2520, 2520, 11, 12

Offset: 1

Reinhard Zumkeller, Jan 22 2003

T(2*n-1,n) = A058312(n-1) for n <= 13.

			Triangle begins:
  1
  2  2
  3  2  3
  4  6  6  4
  5 12  6 12  5
  6 60 12 12 60  6
  7 60 60 12 60 60  7

Cf. A080046.

T(n, 2) = A003418(n-1) for n > 2.

T(n,k) = if ((k==1) || (k==n), n, lcm(T(n-1,k-1), T(n-1,k)));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Apr 25 2018

A213081 Exclusive-or based Pascal triangle, read by rows: T(n,1)=T(n,n)=n and T(n,k) = T(n-1,k-1) XOR T(n-1,k), where XOR is the bitwise exclusive-or operator.

Original entry on oeis.org

1, 2, 2, 3, 0, 3, 4, 3, 3, 4, 5, 7, 0, 7, 5, 6, 2, 7, 7, 2, 6, 7, 4, 5, 0, 5, 4, 7, 8, 3, 1, 5, 5, 1, 3, 8, 9, 11, 2, 4, 0, 4, 2, 11, 9, 10, 2, 9, 6, 4, 4, 6, 9, 2, 10, 11, 8, 11, 15, 2, 0, 2, 15, 11, 8, 11, 12, 3, 3, 4, 13, 2, 2, 13, 4, 3, 3, 12, 13, 15

Offset: 1

Alex Ratushnyak, Jun 04 2012

			Table begins:
   1;
   2,  2;
   3,  0,  3;
   4,  3,  3,  4;
   5,  7,  0,  7,  5;
   6,  2,  7,  7,  2,  6;
   7,  4,  5,  0,  5,  4,  7;
   8,  3,  1,  5,  5,  1,  3,  8;
   9, 11,  2,  4,  0,  4,  2, 11,  9;
  10,  2,  9,  6,  4,  4,  6,  9,  2, 10;
  11,  8, 11, 15,  2,  0,  2, 15, 11,  8, 11;

Cf. A007318 - Pascal's triangle read by rows.
Cf. A051597 - Pascal's triangle, begin and end n-th row with n+1, read by rows.
Cf. A080046 - Multiplicative Pascal triangle, read by rows: T(n,1)=T(n,n)=n and T(n,k) = T(n-1,k-1) * T(n-1,k).

src = [0]*1024
dst = [0]*1024
for i in range(1,39):
    dst[0] = dst[i-1] = i
    for j in range(1,i-1):
        dst[j] = src[j-1]^src[j]
    for j in range(i):
        src[j] = dst[j]
        print(dst[j], end=',')

A317644 Triangle read by rows: multiplicative version of Pascal's triangle except n-th row begins and ends with (n+1)-st prime.

Original entry on oeis.org

2, 3, 3, 5, 9, 5, 7, 45, 45, 7, 11, 315, 2025, 315, 11, 13, 3465, 637875, 637875, 3465, 13, 17, 45045, 2210236875, 406884515625, 2210236875, 45045, 17, 19, 765765, 99560120034375, 899311160300888671875, 899311160300888671875, 99560120034375, 765765, 19, 23, 14549535, 76239655318123171875, 89535527067809533413858673095703125, 808760563041730681160065242862701416015625, 89535527067809533413858673095703125, 76239655318123171875, 14549535, 23

Offset: 0

Philipp O. Tsvetkov, Aug 02 2018

			Triangle begins:
   2;
   3,      3;
   5,      9,      5;
   7,     45,     45,      7;
  11,    315,   2025,    315,     11;
  13,   3465, 637875, 637875,   3465,     13;
  ...
Formatted as a symmetric triangle:
.
                       2
.
                   3       3
.
               5       9       5
.
           7      45      45       7
.
      11      315    2025     315     11
.
  13     3465   637875  637875   3465     13
...

Cf. A000040, A007318, A007949, A028326, A051599, A080046, A112765.

t = {{2}};
Table[AppendTo[
    t, {Prime[i],
      Table[
       t[[i - 1]][[j]]*t[[i - 1]][[j + 1]], {j,
        1, (t[[i - 1]] // Length) - 1}], Prime[i]} // Flatten], {i, 2, 10}] //
   Last // Flatten
t={}; Do[r={}; Do[If[k==0||k==n, m=Prime[n + 1], m=t[[n, k]]t[[n, k + 1]]]; r=AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t (* Vincenzo Librandi, Sep 03 2018 *)

From Rémy Sigrist, Sep 02 2018: (Start)
A007949(T(n+1, k+1)) = A028326(n, k) for any n >= 0 and k = 0..n.
A112765(T(n+1, k+1)) = A007318(n, k) for any n > 0 and k = 0..n.
(End)

cp's OEIS Frontend

A082611 a(n) = Sum_{i=1..k} T(i,k)/(i*(k-i+1)), (k>=1, 1<=n<=k), where T(,) is given in A080046.

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A052849 a(0) = 0; a(n) = 2*n! (n >= 1).

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A079542 Triangular array: T(n,1) = T(n,n) = n and T(n,k) = lcm(T(n-1,k-1), T(n-1,k)) for 1 < k < n.

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A213081 Exclusive-or based Pascal triangle, read by rows: T(n,1)=T(n,n)=n and T(n,k) = T(n-1,k-1) XOR T(n-1,k), where XOR is the bitwise exclusive-or operator.

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A317644 Triangle read by rows: multiplicative version of Pascal's triangle except n-th row begins and ends with (n+1)-st prime.

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Formula