A052578 -id:A052578 - cp's OEIS frontend

A052648 Expansion of e.g.f. 5*x/(1-x).

0, 5, 10, 30, 120, 600, 3600, 25200, 201600, 1814400, 18144000, 199584000, 2395008000, 31135104000, 435891456000, 6538371840000, 104613949440000, 1778437140480000, 32011868528640000, 608225502044160000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 595
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Cf. A000142, A052849 (k=2), A052560 (k=3), A052578 (k=4).

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

With[{nn=20},CoefficientList[Series[(5x)/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 01 2016 *)

E.g.f.: 5*x/(1-x)
Recurrence: {a(0)=0, (-1-n)*a(n)+a(n+1)=0, a(1)=5}
a(n) = 5*n!, n>0.

A159038 a(n) = 8 * n!.

Original entry on oeis.org

8, 16, 48, 192, 960, 5760, 40320, 322560, 2903040, 29030400, 319334400, 3832012800, 49816166400, 697426329600, 10461394944000, 167382319104000, 2845499424768000, 51218989645824000, 973160803270656000

Offset: 1

Zerinvary Lajos, Apr 03 2009

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

Cf. A000142, A052560, A052578, A052648, A052849, A062098.

A159038 := proc(n)
    8*n! ;
end proc:
seq(A159038(n),n=1..10) ; # R. J. Mathar, Sep 30 2013

a(n) = 8 * A000142(n) for n > 0.

A298881 a(0) = 0; for n>0, a(n) = 6*n!.

Original entry on oeis.org

0, 6, 12, 36, 144, 720, 4320, 30240, 241920, 2177280, 21772800, 239500800, 2874009600, 37362124800, 523069747200, 7846046208000, 125536739328000, 2134124568576000, 38414242234368000, 729870602452992000, 14597412049059840000, 306545653030256640000

Offset: 0

Vincenzo Librandi, Feb 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A274266.

Cf. sequences of the type k*n!: A000142 (k=1), A052849 (k=2), A052560 (k=3), A052578 (k=4), A052648 (k=5), this sequence (k=6), A062098 (k=7), A159038 (k=8), A174183 (k=10).

Concatenation([0], List([1..25], n -> 6*Factorial(n))); # Bruno Berselli, Feb 13 2018

[n eq 0 select 0 else 6*Factorial(n): n in [0..25]];

Mathematica
```
Join[{0}, 6 Range[25]!]
```

a(n) = if (n, 6*n!, 0); \\ Michel Marcus, Feb 15 2018

E.g.f.: 6*x/(1-x).

a(n) = n*a(n-1) = 6*A000142(n) for n>0.

Edited by Bruno Berselli, Feb 13 2018

A384494 Triangle read by rows: T(n, k) = (-1)^k(k+1)(n+1-k)!, n >= 0, k = 0..n.

Original entry on oeis.org

1, 2, -2, 6, -4, 3, 24, -12, 6, -4, 120, -48, 18, -8, 5, 720, -240, 72, -24, 10, -6, 5040, -1440, 360, -96, 30, -12, 7, 40320, -10080, 2160, -480, 120, -36, 14, -8, 362880, -80640, 15120, -2880, 600, -144, 42, -16, 9, 3628800, -725760, 120960, -20160, 3600, -720, 168, -48, 18, -10

Offset: 0

Wolfdieter Lang, May 31 2025

This triangle, written as (infinite) square matrix MT with vanishing upper diagonals 0, together with the Riordan triangle A104698, written also as such a square matrix MR, appears in the double sum formula for the number of certain restricted permutations given in A086852(n), as diagonal sequence A086852(n+2) = (2*MR*MT^t)_{n,n}, for n >=0, where t indicates matrix transpositon.

			The triangle T begins:
  n\k        0        1       2       3     4     5   6   7  8    9 ...
  ---------------------------------------------------------------------
  0:         1
  1:         2       -2
  2:         6       -4       3
  3:        24      -12       6      -4
  4:       120      -48      18      -8     5
  5:       720     -240      72     -24    10    -6
  6:      5040    -1440     360     -96    30   -12   7
  7:     40320   -10080    2160    -480    12   -36  14  -8
  8:    362880   -80640   15120   -2880   600  -144  42 -16  9
  9:   3628800  -725760  120960  -20160  3600  -720 168 -48 18 -10
  ...

Cf. A104698, A086852, A086852.

Column sequences: A000142(n+1), -A052849, A052560(n-1), -A052578(n-2), A052648(n-3), -A298881(n-4), A062098(n-5), -A159038(n-6), ...

Table[(-1)^k * (k+1) * (n+1-k)!, {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 31 2025 *)

T(n, k) = (-1)^k*(k+1)*(n+1-k)!, for n >= 0 and k = 0, 1, ..., n.

O.g.f. of row polynomials P(n, y) := Sum_{k=0..n} T(n, k) y^k: G(x, y) = ((N(x) - 1)/x) * (1/(1 + y*x)^2), with N(x) = hypergeometric([1,1], [], x), the o.g.f. of {n!}_{n>=0} (see A000142).

A117826 First four terms of the sequence are doubled, then those numbers are tripled and then those numbers are quadrupled, etc.

Original entry on oeis.org

1, 2, 3, 4, 2, 4, 6, 8, 6, 12, 18, 24, 24, 48, 72, 96, 120, 240, 360, 480, 720, 1440, 2160, 2880, 5040, 10080, 15120, 20160, 40320, 80640, 120960, 161280, 362880, 725760, 1088640, 1451520, 3628800, 7257600, 10886400, 14515200, 39916800, 79833600

Offset: 1

Robert G. Wilson v, Apr 25 2006

nonn

Cf. A000142, A115425.

Table[n!{1, 2, 3, 4}, {n, 11}] // Flatten

a(4n-3) = A000142(n). a(4n-2)=2*A000142(n)=A052849(n), a(4n-1)=3*A000142(n)=A052560(n), a(4n)=4*A000142(n)=2*A052849(n)=A052578(n).

cp's OEIS Frontend

A052648 Expansion of e.g.f. 5*x/(1-x).

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Maple

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A159038 a(n) = 8 * n!.

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A298881 a(0) = 0; for n>0, a(n) = 6*n!.

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A384494 Triangle read by rows: T(n, k) = (-1)^k*(k+1)*(n+1-k)!, n >= 0, k = 0..n.

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A117826 First four terms of the sequence are doubled, then those numbers are tripled and then those numbers are quadrupled, etc.

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A384494 Triangle read by rows: T(n, k) = (-1)^k(k+1)(n+1-k)!, n >= 0, k = 0..n.