A000442 -id:A000442 - cp's OEIS frontend

A061719 a(n) = Product_{k=0...n} (k!^3).

1, 1, 8, 1728, 23887872, 41278242816000, 15407021574586368000000, 1972469516114225950359552000000000, 129292064547357027522197559428775936000000000000

Offset: 0

Jason Earls, Jun 20 2001

Harry J. Smith, Table of n, a(n) for n=0..27

Cf. A000178.

Table[Product[k!^3, {k, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 23 2023 *)

for(n=0,11,print(prod(k=1,n,factorial(k)^3)))

{ for (n=0, 27, write("b061719.txt", n, " ", prod(k=2, n, k!^3)) ) } \\ Harry J. Smith, Jul 26 2009

a(n) = a(n-1)*A000442(n). - R. J. Mathar, Sep 26 2020
From Vaclav Kotesovec, Nov 23 2023: (Start)
a(n) = A000178(n)^3.
a(n) ~ (2*Pi)^(3*n/2 + 3/2) * n^(3*n^2/2 + 3*n + 5/4) / (A^3 * exp(9*n^2/4 + 3*n - 1/4)), where A is the Glaisher-Kinkelin constant A074962. (End)

Terms corrected according to Jason Earls's instructions by Harry J. Smith, Jul 26 2009

A334394 Triangle read by rows: T(n,k) is the number of ordered triples of n-permutations with exactly k common descents, n>=0, 0<=k<=max(0,n-1).

Original entry on oeis.org

1, 1, 7, 1, 163, 52, 1, 8983, 4499, 341, 1, 966751, 660746, 98256, 2246, 1, 179781181, 155729277, 35677082, 2045282, 15177, 1, 53090086057, 55690144728, 17446464519, 1754605504, 42658239, 104952, 1, 23402291822743, 28825420903351, 11518335730323, 1717307782339, 84058424389, 905365701, 739153, 1

Offset: 0

Geoffrey Critzer, Apr 26 2020

An ordered triple of n-permutations ( (a_1,a_2,...,a_n),(b_1,b_2,...,b_n),(c_1,c_2,...,c_n) ) has a common descent at position i, 1<=i<=n-1, if a_i > a_i+1, b_i > b_i+1 and c_i > c_i+1.

			Triangle begins:
       1;
       1;
       7,      1;
     163,     52,     1;
    8983,   4499,   341,   1;
  966751, 660746, 98256, 2246, 1;
  ...

R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, example 3.18.3e, page 366.

Alois P. Heinz, Rows n = 0..30, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 209.

Cf. A192721, A008292, A212856 (column k=0), A000442 (row sums).

T:= (n, k)-> n!^3*coeff(series(coeff(series((y-1)/(y-add((x*
    (y-1))^j/j!^3, j=0..n)), y, k+1), y, k), x, n+1), x, n):
seq(seq(T(n,k), k=0..max(0, n-1)), n=0..10);  # Alois P. Heinz, Apr 28 2020

nn = 6; e3[x_] := Sum[x^n/n!^3, {n, 0, nn}];Drop[Map[Select[#, # > 0 &] &,
   Table[n!^3, {n, 0, nn}] CoefficientList[Series[(y - 1)/(y - e3[x (y - 1)]), {x, 0, nn}], {x, y}]], 1] // Grid

Sum_{n>=0} Sum_{k>=0} T(n,k)*y^k*x^n/n!^3 = (y-1)/(y-f(x*(y-1))) where f(z) = Sum_{n>=0} z^n/n!^3.

A384044 a(n) = [x^n] Product_{k=1..n} (1 + k^3x) / (1 - k^3x).

Original entry on oeis.org

1, 2, 162, 75672, 104312000, 317309605650, 1803288012589602, 17180843554017736544, 254292459616733559570432, 5525508321588276184345621650, 168733575675064578625834983478850, 6994229599670887851052241626545021912, 382562895157136117988572795915676719695680

Offset: 0

Vaclav Kotesovec, May 18 2025

nonn

Cf. A000442, A351800.

Cf. A350366, A384043, A351764.

Table[SeriesCoefficient[Product[(1+k^3*x)/(1-k^3*x), {k, 1, n}], {x, 0, n}], {n, 0, 12}]

a(n) ~ c * d^n * n!^3 / n^2, where d = 37.604795475701444958019770120055586495991039059348094619704... and c = 0.063895861310548119570865800164582089372152350471371583403...

A154226 List of pairs: {(n*(n+1)/2)^2, (n!)^3}.

Original entry on oeis.org

0, 1, 1, 1, 9, 8, 36, 216, 100, 13824, 225, 1728000, 441, 373248000, 784, 128024064000, 1296, 65548320768000, 2025, 47784725839872000, 3025, 47784725839872000000, 4356, 63601470092869632000000, 6084

Offset: 0

Roger L. Bagula, Jan 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000442, A000537.

&cat[[(n*(n+1)/2)^2, Factorial(n)^3]: n in [0..15]]; // Vincenzo Librandi, Sep 07 2016

a[0] = 0; a[n_] := a[n] = n^3 + a[n - 1];
b[0] = 1; b[n_] := b[n] = n^3*b[n - 1];
Flatten[Table[{a[n], b[n]}, {n, 0, 15}]]

a(2n+1) = A000442(n) = (n!)^3.

a(2n) = A000537(n) = (n*(n+1)/2)^2.

A287697 Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287698.

Original entry on oeis.org

1, 0, 1, 0, 1, 7, 0, 1, 52, 163, 0, 1, 341, 4499, 8983, 0, 1, 2246, 98256, 660746, 966751, 0, 1, 15177, 2045282, 35677082, 155729277, 179781181, 0, 1, 104952, 42658239, 1754605504, 17446464519, 55690144728, 53090086057

Offset: 0

Peter Luschny, May 30 2017

			Triangle starts:
0: [1]
1: [0, 1]
2: [0, 1,      7]
3: [0, 1,     52,      163]
4: [0, 1,    341,     4499,       8983]
5: [0, 1,   2246,    98256,     660746,      966751]
6: [0, 1,  15177,  2045282,   35677082,   155729277,   179781181]
7: [0, 1, 104952, 42658239, 1754605504, 17446464519, 55690144728, 53090086057]
...
Let q4(x) = (x + 341*x^2 + 4499*x^3 + 8983*x^4) / (1-x)^5 then the coefficients of the series expansion of q4 are column 4 of A287698.

Cf. A000442, A212856, A287696, A287698.

A287697_row := n -> Delta(A287696_poly(n), n): # Delta defined in A287315.
for n from 0 to 9 do A287697_row(n) od;
A287697_eulerian := (n,x) -> add(A287697_row(n)[k+1]*x^k,k=0..n)/(1-x)^(n+1):
for n from 0 to 4 do A287697_eulerian(n,x) od;

T(n,n) = A212856(n).

Sum_{k=0..n} T(n,k) = A000442(n).

A369698 AGM transform of positive cubes.

Original entry on oeis.org

0, 49, 40824, 96461056, 571250390625, 7338413252698641, 181953686508203782144, 7957561391610438862503936, 572547082070592542500791107625, 64157961305703333114506988525390625, 10714350425499230222239742740718898118656, 2571996060859292513876561308464753498396819456

Offset: 1

Paolo Xausa, Jan 29 2024

nonn

See A368366 for the description of the AGM transform.

Paolo Xausa, Table of n, a(n) for n = 1..120

Cf. A368366, A368368.

Cf. A000312, A000442, A000537.

A369698[n_] := (n*(n+1)/2)^(2*n) - n^n*n!^3; Array[A369698, 15]

from math import factorial
def A369698(n): return ((n*(n+1))**(m:=n<<1)>>m) - n**n*factorial(n)**3 # Chai Wah Wu, Jan 29 2024

a(n) = A000537(n)^n - A000312(n)*A000442(n).

A381160 a(n) is the permanent of the n X n matrix whose element (i,j) is equal to A008277(i+3, j) with 1 <= i,j <= n.

Original entry on oeis.org

1, 1, 22, 3206, 1902936, 3504528354, 16660734321540, 179059038168086056, 3938830136216956996632, 164125096331945477980176920, 12173562237817299484378342192768, 1527294306324982018922212102518520032, 310564445230567070838152555220146533261496, 98712056006032672983172826864304778359411112064

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

			a(3) = 3206:
  [1,  7,  6]
  [1, 15, 25]
  [1, 31, 90]

Cf. A000442 (determinant), A008277, A381166.

a[n_]:=Permanent[Table[StirlingS2[i+3,j],{i,n},{j,n}]]; Join[{1},Array[a,13]]

a(n) = matpermanent(matrix(n, n, i, j, stirling(i+3,j,2))); \\ Michel Marcus, Feb 16 2025

A127488 a(n) = (n^2)!/(2*(n!)).

Original entry on oeis.org

6, 30240, 435891456000, 64630041847212441600000, 258328699159653623241666283438080000000

Offset: 2

Artur Jasinski, Jan 16 2007

nonn

Cf. A000142, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134374, A134374

Mathematica
```
Table[(n^2)!/(2(n!)), {n, 2, 6}]
```

a(n) ~ n^(2*n^2 - n + 1/2) / (2 * exp(n*(n-1))). - Vaclav Kotesovec, Oct 26 2017

A380318 Product of the first n perfect powers (A001597).

Original entry on oeis.org

1, 1, 4, 32, 288, 4608, 115200, 3110400, 99532800, 3583180800, 175575859200, 11236854988800, 910185254092800, 91018525409280000, 11013241574522880000, 1376655196815360000000, 176211865192366080000000, 25374508587700715520000000, 4288291951321420922880000000, 840505222458998500884480000000, 181549128051143676191047680000000

Offset: 0

Ilya Gutkovskiy, Jan 20 2025

nonn

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A000442, A001044, A001597, A002110, A024923, A036691, A076408, A111059.

Join[{1},FoldList[Times,Join[{1},Select[Range[250],GCD@@FactorInteger[#][[All,2]]>1&]]]] (* Harvey P. Dale, May 03 2025 *)

A386405 Decimal expansion of Product_{k>=0} (1 + 1/k!^3).

Original entry on oeis.org

4, 5, 2, 1, 1, 6, 2, 9, 8, 9, 7, 6, 8, 2, 1, 8, 4, 8, 1, 0, 2, 5, 3, 5, 9, 9, 3, 6, 0, 6, 2, 8, 3, 4, 8, 8, 6, 9, 7, 2, 4, 1, 5, 0, 1, 9, 8, 7, 1, 8, 8, 3, 1, 9, 2, 0, 7, 0, 8, 9, 9, 0, 9, 5, 5, 1, 6, 4, 3, 5, 0, 1, 5, 6, 2, 2, 5, 4, 5, 8, 2, 9, 4, 8, 0, 4, 2, 4, 9, 5, 6, 7, 6, 4, 3, 6, 6, 4, 5, 6, 9, 4, 1, 5, 6

Offset: 1

Kelvin Voskuijl, Aug 20 2025

			4.52116298976821848102535993606283488697241501987188319...

Cf. A000442, A238695, A384510.

PARI
```
prodinf(k=0, 1 + 1/k!^3)
```

cp's OEIS Frontend

A061719 a(n) = Product_{k=0...n} (k!^3).

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A334394 Triangle read by rows: T(n,k) is the number of ordered triples of n-permutations with exactly k common descents, n>=0, 0<=k<=max(0,n-1).

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A384044 a(n) = [x^n] Product_{k=1..n} (1 + k^3*x) / (1 - k^3*x).

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A154226 List of pairs: {(n*(n+1)/2)^2, (n!)^3}.

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Magma

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A287697 Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287698.

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A369698 AGM transform of positive cubes.

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A381160 a(n) is the permanent of the n X n matrix whose element (i,j) is equal to A008277(i+3, j) with 1 <= i,j <= n.

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A127488 a(n) = (n^2)!/(2*(n!)).

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A384044 a(n) = [x^n] Product_{k=1..n} (1 + k^3x) / (1 - k^3x).