A052560 -id:A052560 - cp's OEIS frontend

A276955 Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

1, 2, 3, 6, 8, 4, 24, 30, 12, 5, 120, 144, 48, 14, 7, 720, 840, 240, 54, 26, 9, 5040, 5760, 1440, 264, 126, 32, 10, 40320, 45360, 10080, 1560, 744, 150, 36, 11, 362880, 403200, 80640, 10800, 5160, 864, 168, 38, 13, 3628800, 3991680, 725760, 85680, 41040, 5880, 960, 174, 50, 15, 39916800, 43545600, 7257600, 766080, 367920, 46080, 6480, 984, 246, 56, 16

Offset: 1

Antti Karttunen, Sep 22 2016

The square array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

When viewed in factorial base (A007623) the terms on each row start all with the same prefix, but with an increasing number of zeros appended to the end. For example, for row 8 (A001344 from a(1)=11 onward), the terms written in factorial base look as: 121, 1210, 12100, 121000, ...

			The top left {1..9} x {1..18} corner of the array:
   1,  2,   6,   24,   120,    720,    5040,    40320,    362880
   3,  8,  30,  144,   840,   5760,   45360,   403200,   3991680
   4, 12,  48,  240,  1440,  10080,   80640,   725760,   7257600
   5, 14,  54,  264,  1560,  10800,   85680,   766080,   7620480
   7, 26, 126,  744,  5160,  41040,  367920,  3669120,  40279680
   9, 32, 150,  864,  5880,  46080,  408240,  4032000,  43908480
  10, 36, 168,  960,  6480,  50400,  443520,  4354560,  47174400
  11, 38, 174,  984,  6600,  51120,  448560,  4394880,  47537280
  13, 50, 246, 1464, 10200,  81360,  730800,  7297920,  80196480
  15, 56, 270, 1584, 10920,  86400,  771120,  7660800,  83825280
  16, 60, 288, 1680, 11520,  90720,  806400,  7983360,  87091200
  17, 62, 294, 1704, 11640,  91440,  811440,  8023680,  87454080
  18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400
  19, 74, 366, 2184, 15240, 121680, 1093680, 10926720, 120113280
  20, 78, 384, 2280, 15840, 126000, 1128960, 11249280, 123379200
  21, 80, 390, 2304, 15960, 126720, 1134000, 11289600, 123742080
  22, 84, 408, 2400, 16560, 131040, 1169280, 11612160, 127008000
  23, 86, 414, 2424, 16680, 131760, 1174320, 11652480, 127370880

Inverse permutation: A276956.
Transpose: A276953.
Cf. A276949 (index of column where n appears), A276951 (index of row).
Cf. A153880.
Columns 1-3: A273670, A276932, A276933.
The following lists some of the rows that have their own entries. Pattern present in the factorial base expansion of the terms on that row is given in double quotes:
Row 1: A000142 (from a(1)=1, "1" onward),
Row 2: A001048 (from a(2)=3, "11" onward),
Row 3: A052849 (from a(2)=4, "20" onward).
Row 4: A052649 (from a(1)=5, "21" onward).
Row 5: A108217 (from a(3)=7, "101" onward).
Row 6: A054119 (from a(3)=9, "111" onward).
Row 7: A052572 (from a(2)=10, "120" onward).
Row 8: A001344 (from a(1)=11, "121" onward).
Row 13: A052560 (from a(3)=18, "300" onward).
Row 16: A225658 (from a(1)=21, "311" onward).
Row 20: A276940 (from a(3) = 27, "1011" onward).
Related or similar permutations: A257505, A275848, A273666.
Cf. also arrays A276617, A276588 & A276945.

(define (A276955 n) (A276955bi (A002260 n) (A004736 n)))
(define (A276955bi row col) (if (= 1 col) (A273670 (- row 1)) (A153880 (A276955bi row (- col 1)))))

A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1))
As a composition of other permutations:
a(n) = A275848(A257505(n)).

A052578 a(0) = 0, a(n) = 4*n! for n > 0.

Original entry on oeis.org

0, 4, 8, 24, 96, 480, 2880, 20160, 161280, 1451520, 14515200, 159667200, 1916006400, 24908083200, 348713164800, 5230697472000, 83691159552000, 1422749712384000, 25609494822912000, 486580401635328000, 9731608032706560000, 204363768686837760000

Offset: 0

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

Vincenzo Librandi, Table of n, a(n) for n = 0..447
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 522

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

[0] cat [4*Factorial(n): n in [1..25]]; // Vincenzo Librandi, Feb 13 2018

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

Join[{0},4Range[25]!]  (* Harvey P. Dale, Mar 12 2011 *)

a(n)=if(n,4*n!,0) \\ Charles R Greathouse IV, Nov 20 2011

E.g.f.: -4*x/(-1+x)

Recurrence: {a(0)=0, (-1-n)*a(n)+a(n+1), a(1)=4}

A083746 a(1) = 1, a(2) = 2; for n>2, a(n) = 3(n-2)(n-2)!.

Original entry on oeis.org

1, 2, 3, 12, 54, 288, 1800, 12960, 105840, 967680, 9797760, 108864000, 1317254400, 17244057600, 242853811200, 3661488230400, 58845346560000, 1004293914624000, 18140058832896000, 345728180109312000, 6933770723303424000

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

nonn

a(1) = 1, a(2) = 2, define S(k) = sum of all the terms other than a(k) k < n. a(n) = Sum_{k=1..n-1} S(k).

			a(4) = {a(1) + a(2)} + {a(1) +a(3)} + {a(2) + a(3)} = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Cf. A001563, A052673, A129379.

[n le 2 select n else 3*(n-2)*Factorial(n-2): n in [1..40]]; // G. C. Greubel, Feb 03 2024

a := proc(n) option remember: if n=1 then RETURN(1) fi: if n=2 then RETURN(2) fi: 3*(n-2)*(n-2)! end: for n from 1 to 40 do printf(`%d,`,a(n)) od: # James Sellers, May 19 2003

Join[{1,2},Table[3n n!,{n,20}]] (* Harvey P. Dale, Feb 27 2012 *)

[1,2]+[3*(n-2)*factorial(n-2) for n in range(3, 41)] # G. C. Greubel, Feb 03 2024

a(n) = (n-2)*Sum_{j=1..n-1} a(j).
E.g.f.: 3*(x-2)*log(1-x) - 5*x + x^2. - Vladeta Jovovic, May 06 2003
From Reinhard Zumkeller, Apr 14 2007: (Start)
Sum_{k=1..n} a(k) = A052560(n-1) for n > 1.
a(n) = A052673(n-2) for n > 2. (End)

Simpler description from Vladeta Jovovic, May 06 2003

More terms from James Sellers, May 19 2003

A110141 Triangle, read by rows, where row n lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k).

Original entry on oeis.org

1, 1, 2, 2, 6, 2, 3, 24, 4, 3, 8, 4, 120, 12, 6, 8, 4, 6, 5, 720, 48, 18, 16, 8, 6, 5, 48, 8, 18, 6, 5040, 240, 72, 48, 24, 12, 10, 48, 8, 18, 6, 24, 10, 12, 7, 40320, 1440, 360, 192, 96, 36, 30, 96, 16, 36, 12, 24, 10, 12, 7, 384, 32, 36, 12, 15, 32, 8, 362880, 10080, 2160, 960

Offset: 0

Paul D. Hanna, Jul 13 2005

Row n starts with n!, after which the following pattern holds. When terms of row n are divided by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1, the result is the initial terms of A110142. E.g., row 6 is: {720,48,18,16,8,6,5,48,8,18,6}; divide by respective factorials: {6!,4!,3!,2!,2!,1!,1!,0!,0!,0!,0!} with {4!,3!,2!,1!,0!} respectively occurring {1,1,2,2,4} times (A002865), yields the initial terms of A110142: {1,2,3,8,4,6,5,48,8,18,6}.
The term of the sequence corresponding to the product c_1^{n_1}c_2^{n_2}...c_k^{n_k} is equal to the number of elements in the centralizer of a permutation of n_1+2n_2+...+kn_k elements whose cycle type is 1^{n_1}2^{n_2}...k^{n^k}. (This fact is very standard, in particular, for the theory of symmetric functions.) - Vladimir Dotsenko, Apr 19 2009
Multiplying the values of row n by the corresponding values in row n of A102189, one obtains n!. - Jaimal Ichharam, Aug 06 2015
a(n,k) is the number of permutations in S_n that commute with a permutation having cycle type "k".  Here, the cycle type of an n-permutation pi is the vector (i_1,...,i_n) where i_j is the number of cycles in pi of length j.  These A000041(n) vectors can be ordered in reverse lexicographic order.  The k-th cycle type is the k-th vector in this ordering. - Geoffrey Critzer, Jan 18 2019

			Coefficients [x^n] exp(c1*x + (c2/2)*x^2 + (c3/3)*x^3 + ...) begin:
[x^0]: 1;
[x^1]: 1*c1;
[x^2]: (1/2)*c1^2 + (1/2)*c2;
[x^3]: (1/6)*c1^3 + (1/2)*c1*c2 + (1/3)*c3;
[x^4]: (1/24)*c1^4 + (1/4)*c1^2*c2 + (1/3)*c1*c3 + (1/8)*c2^2 + (1/4)*c4;
[x^5]: (1/120)*c1^5 + (1/12)*c1^3*c2 + (1/6)*c1^2*c3 + (1/8)*c1*c2^2 + (1/4)*c1*c4 + (1/6)*c2*c3 + (1/5)*c5;
[x^6]: (1/720)*c1^6 + (1/48)*c1^4*c2 + (1/18)*c1^3*c3 + (1/16)*c1^2*c2^2 + (1/8)*c1^2*c4 + (1/6)*c1*c2*c3 + (1/5)*c1*c5 + (1/48)*c2^3 + (1/8)*c2*c4 + (1/18)*c3^2 + (1/6)*c6;
forming this triangle of unit fraction coefficients:
1;
1;
2,2;
6,2,3;
24,4,3,8,4;
120,12,6,8,4,6,5;
720,48,18,16,8,6,5,48,8,18,6;
5040,240,72,48,24,12,10,48,8,18,6,24,10,12,7;
40320,1440,360,192,96,36,30,96,16,36,12,24,10,12,7,384,32,36,12,15,32,8;
362880,10080,2160,960,480,144,120,288,48,108,36,48,20,24,14,384,32,36,12,15,32,8,144,40,24,14,162,18,20,9; ...

Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford University Press, 1995. [From Vladimir Dotsenko, Apr 19 2009]

Cf. A000041, A002865, A102189, A110142, A110143 (row sums).

First, second and third entries of each row are given (up to an offset) by A000142, A052849, and A052560 respectively. - Vladimir Dotsenko, Apr 19 2009

Table[n!/CoefficientRules[n! CycleIndex[SymmetricGroup[n], s]][[All, 2]], {n, 1, 8}] // Grid (* Geoffrey Critzer, Jan 18 2019 *)

Number of terms in row n is A000041(n) (partition numbers). The unit fractions of each row sum to unity: Sum_{k=1..A000041(n)} 1/T(n, k) = 1.

a(n,k) = n!/A181897(n,k). - Geoffrey Critzer, Jan 18 2019

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

Original entry on oeis.org

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.

A129380 Partial sums of A129379.

Original entry on oeis.org

1, 3, 6, 12, 48, 336, 3696, 59136, 1300992, 37728768, 1395964416, 64214363136, 3596004335616, 240932290486272, 19033650948415488, 1751095887254224896, 185616164048947838976, 22459555849922688516096

Offset: 1

Reinhard Zumkeller, Apr 14 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..270

Cf. A000217, A052560, A129379.

A129380:= func< n | n le 2 select 2*n-1 else (3/2^(n-2))*(&*[k^2-k+2: k in [0..n-2]])  >;
[A129380(n): n in [1..40]]; // G. C. Greubel, Feb 03 2024

a[n_]:= a[n]= If[n<4, Binomial[n+1,2], (n^2-5*n+8)*a[n-1]/2];
Table[a[n], {n,40}] (* G. C. Greubel, Feb 03 2024 *)
Round[Flatten[{{1, 3}, Table[(3*2^(3-n) * Cosh[Sqrt[7]*Pi/2] * Gamma[n - 3/2 - I*Sqrt[7]/2] * Gamma[n - 3/2 + I*Sqrt[7]/2])/Pi, {n, 3, 20}]}]] (* Vaclav Kotesovec, Feb 03 2024 *)

def A129380(n): return 2*n-1 if n<3 else 3*product(j^2-j+2 for j in range(n-1))//2^(n-2)
[A129380(n) for n in range(1, 41)] # G. C. Greubel, Feb 03 2024

a(n) = A129379(n+1)/A000217(n-2) for n>2.
a(n) = a(n-1) + A129379(n) for n > 1, a(1) = 1.
From G. C. Greubel, Feb 03 2024: (Start)
a(n) = (6/2^(n-3))*|Pochhammer((3+i*sqrt(7))/2, n-3)|^2, for n > 2.
a(n) = (3/2^(n-3))*Product_{k=0..n-2} (k^2 - k + 2), for n > 2.
a(n) = (1/2)*(n^2 - 5*n + 8)*a(n-1), with a(1) = 1, a(2) = 3, a(3) = 6. (End)
From Vaclav Kotesovec, Feb 03 2024: (Start)
For n>=3, a(n) = 3 * cosh(sqrt(7)*Pi/2) * 2^(3-n) * Gamma(n - 3/2 - i*sqrt(7)/2) * Gamma(n - 3/2 + i*sqrt(7)/2)/Pi, where i is the imaginary unit.
a(n) ~ 3 * cosh(sqrt(7)*Pi/2) * n^(2*n-4) / (2^(n-4) * exp(2*n)). (End)

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

A324225 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 6, 4, 2, 6, 12, 18, 24, 18, 12, 6, 24, 48, 72, 96, 120, 96, 72, 48, 24, 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120, 720, 1440, 2160, 2880, 3600, 4320, 5040, 4320, 3600, 2880, 2160, 1440, 720, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 35280, 30240, 25200, 20160, 15120, 10080, 5040

Offset: 1

Alois P. Heinz, Feb 18 2019

T(n,k) is the number of occurrences of k in all (signed) displacement lists [p(i)-i, i=1..n] of permutations p of [n].

			The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement lists [p(i)-i, i=1..3]: [0,0,0], [0,1,-1], [1,-1,0], [1,1,-2], [2,-1,-1], [2,0,-2], representing the indices of falling diagonals of 1's in the permutation matrices
  [1    ]  [1    ]  [  1  ]  [  1  ]  [    1]  [    1]
  [  1  ]  [    1]  [1    ]  [    1]  [1    ]  [  1  ]
  [    1]  [  1  ]  [    1]  [1    ]  [  1  ]  [1    ] , respectively. Indices -2 and 2 occur twice, -1 and 1 occur four times, and 0 occurs six times. So row n=3 is [2, 4, 6, 4, 2].
Triangle T(n,k) begins:
  :                             1                           ;
  :                        1,   2,   1                      ;
  :                   2,   4,   6,   4,   2                 ;
  :              6,  12,  18,  24,  18,  12,   6            ;
  :        24,  48,  72,  96, 120,  96,  72,  48,  24       ;
  :  120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120  ;

Alois P. Heinz, Rows n = 1..100, flattened
Nadir Samos Sáenz de Buruaga, Rafał Bistroń, Marcin Rudziński, Rodrigo Miguel Chinita Pereira, Karol Życzkowski, and Pedro Ribeiro, Fidelity decay and error accumulation in quantum volume circuits, arXiv:2404.11444 [quant-ph], 2024. See p. 18.
Wikipedia, Permutation
Wikipedia, Permutation matrix

Columns k=0-6 give (offsets may differ): A000142, A001563, A062119, A052571, A052520, A282822, A052521.
Row sums give A001563.
T(n+1,n) gives A000142.
T(n+1,n-1) gives A052849.
T(n+1,n-2) gives A052560 for n>1.
Cf. A152883 (right half of this triangle without center column), A162608 (left half of this triangle), A306461, A324224.
Cf. A001710.

b:= proc(s, c) option remember; (n-> `if`(n=0, c,
      add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t

T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)

T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)! if t < n with t = |k|, T(n,k) = 0 otherwise.
T(n,k) = |k|! * A324224(n,k).
E.g.f. of column k: x^t/t * hypergeom([2, t], [t+1], x) with t = |k|+1.
|T(n,k)-T(n,k-1)| = (n-1)! for k = 1-n..n.
Sum_{k=0..n-1} T(n,k) = A001710(n+1).

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).

cp's OEIS Frontend

A276955 Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

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A052578 a(0) = 0, a(n) = 4*n! for n > 0.

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A083746 a(1) = 1, a(2) = 2; for n>2, a(n) = 3*(n-2)*(n-2)!.

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A110141 Triangle, read by rows, where row n lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k).

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

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A129380 Partial sums of A129379.

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A159038 a(n) = 8 * n!.

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A083746 a(1) = 1, a(2) = 2; for n>2, a(n) = 3(n-2)(n-2)!.

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