A000442 -id:A000442 - cp's OEIS frontend

A225816 Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 4, 1, 1, 1, 24, 36, 8, 1, 1, 1, 120, 576, 216, 16, 1, 1, 1, 720, 14400, 13824, 1296, 32, 1, 1, 1, 5040, 518400, 1728000, 331776, 7776, 64, 1, 1, 1, 40320, 25401600, 373248000, 207360000, 7962624, 46656, 128, 1, 1

Offset: 0

Alois P. Heinz, Jul 29 2013

A(n,k) is the determinant of the k X k matrix M = [Stirling2(n+i,j)] for 1<=i,j<=k.  A(2,3) = det([1,3,1; 1,7,6; 1,15,25]) = 36.
A(n,k) is the determinant of the symmetric k X k matrix M = [sigma_n(gcd(i,j))] for 1<=i,j<=k.  A(2,3) = det([1,1,1; 1,5,1; 1,1,10]) = 36.
A(n,k) is (-1)^(n*k) times the determinant of the n X n matrix M = [Stirling1(k+i,j)] for 1<=i,j<=n.  A(2,3) = (-1)^(2+3) * det([-6,11; 24,-50]) = 36.
A(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_i-p_j) <= 1 for 1<=i,j<=k.  A(2,3) = 36:
          (1,2,2)-(1,1,2)         (0,1,1)-(0,0,1)
         /       X       \       /       X       \
  (2,2,2)-(2,1,2) (1,2,1)-(1,1,1)-(1,0,1) (0,1,0)-(0,0,0).
         \       X       /       \       X       /
          (2,2,1) (2,1,1)         (1,1,0) (1,0,0)
A(n,k) is the number of set partitions of [k*(n+1)] into k blocks of size n+1 such that the elements of each block are distinct mod n+1.  A(2,3) = 36: 123|456|789, 126|345|789, ..., 189|234|567, 189|246|357.

			Square array A(n,k) begins:
  1, 1,  1,    1,       1,           1, ...
  1, 1,  2,    6,      24,         120, ...
  1, 1,  4,   36,     576,       14400, ...
  1, 1,  8,  216,   13824,     1728000, ...
  1, 1, 16, 1296,  331776,   207360000, ...
  1, 1, 32, 7776, 7962624, 24883200000, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened

Columns k=0+1, 2-4 give: A000012, A000079, A000400, A009968.
Rows n=0-4 give: A000012, A000142, A001044, A000442, A134375.
Main diagonal gives: A036740.
Cf. A008275, A048994, A008277, A048993, A003989, A109004, A109974.

A:= (n, k)-> k!^n:
seq(seq(A(n,d-n), n=0..d), d=0..12);

A(n,k) = (k!)^n.
A(n,k) = k^n * A(n,k-1) for k>0, A(n,0) = 1.
A(n,k) = k!  * A(n-1,k) for n>0, A(0,k) = 1.
G.f. of column k: 1/(1-k!*x).

A134372 a(n) = ((2n)!)^2.

Original entry on oeis.org

1, 4, 576, 518400, 1625702400, 13168189440000, 229442532802560000, 7600054456551997440000, 437763136697395052544000000, 40990389067797283140009984000000, 5919012181389927685417441689600000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A010050, A134366, A134367, A134368, A134369, A134371, A134374, A134375, A334379, A334632.

Table[((2n)!)^(2), {n, 0, 10}]
((2*Range[0,20])!)^2 (* Harvey P. Dale, Jul 14 2011 *)

a(n) = ((2*n)!)^2; \\ Michel Marcus, Nov 16 2020

From Amiram Eldar, Nov 16 2020: (Start)
Sum_{n>=0} 1/a(n) = A334379.
Sum_{n>=0} (-1)^n/a(n) = A334632. (End)

A134370 a(n) = ((2n+1)!)^(n+2).

Original entry on oeis.org

1, 216, 207360000, 3252016064102400000, 2283380023591730815784976384000000, 161469323688736156802100136913438716723200000000000000, 2260697901194635682690248130915498742378267539496871953338204160000000000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A134366, A134367, A134368, A134369, A134371, A134374, A134375.

Mathematica
```
Table[((2n+1)!)^(n + 2), {n, 0, 10}]
```

a(n) ~ 2^(2*(n+1)*(n+2)) * exp(13/24 - 2*n*(n+2)) * n^((n+2)*(4*n+3)/2) * Pi^(n/2 + 1). - Vaclav Kotesovec, Oct 26 2017

Typo in a(6) corrected by Georg Fischer, Apr 10 2024

A134373 a(n) = ((2n)!)^3.

Original entry on oeis.org

1, 8, 13824, 373248000, 65548320768000, 47784725839872000000, 109903340320478724096000000, 662559760549147780765974528000000, 9159226129831418921308831875072000000000, 262435789155225791087396177124997988352000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Christian Elsholtz, Golomb’s Conjecture on Prime Gaps, arXiv:1604.06903 [math.NT], 2016.
Christian Elsholtz, Golomb’s Conjecture on Prime Gaps, The American Mathematical Monthly 124.4 (2017): 365-368.

Cf. A000142, A001044, A000442, A036740, A010050, A134366, A134367, A134368, A134369, A134371, A134373, A134374, A134375.

Table[((2n)!)^(3), {n, 0, 10}]
((2*Range[0, 10])!)^3 (* Harvey P. Dale, Jul 25 2016 *)

[factorial(2*n)**3 for n in range(0,9)] # Stefano Spezia, Apr 22 2025

Definition corrected by Harvey P. Dale, Jul 25 2016

A249677 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^3*x).

Original entry on oeis.org

1, 1, 1, 1, 9, 8, 1, 36, 251, 216, 1, 100, 2555, 16280, 13824, 1, 225, 15055, 335655, 2048824, 1728000, 1, 441, 63655, 3587535, 74550304, 444273984, 373248000, 1, 784, 214918, 25421200, 1305074809, 26015028256, 152759224512, 128024064000, 1, 1296, 616326, 135459216, 14320729209, 694213330464, 13472453691584, 78340747014144, 65548320768000

Offset: 0

Paul D. Hanna, Nov 03 2014

Column 1 forms the squares of the triangular numbers (A000537).
Main diagonal forms the cubes of the factorial numbers (A000442).
Row sums equal Product_{k=1..n} (k^3 + 1) = n!*Product_{k=1..n} (k*(k-1) + 1) = n!*A130032(n).

			Triangle begins:
  1;
  1, 1;
  1, 9, 8;
  1, 36, 251, 216;
  1, 100, 2555, 16280, 13824;
  1, 225, 15055, 335655, 2048824, 1728000;
  1, 441, 63655, 3587535, 74550304, 444273984, 373248000;
  1, 784, 214918, 25421200, 1305074809, 26015028256, 152759224512, 128024064000;
  1, 1296, 616326, 135459216, 14320729209, 694213330464, 13472453691584, 78340747014144, 65548320768000; ...

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A000537, A000442, A008955, A107415, A130032.

{T(n,k)=polcoeff(prod(m=0,n,1 + m^3*x +x*O(x^n)),k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

A381161 a(n) = (10n)!/((n!)^3(2n)!(5*n)!).

Original entry on oeis.org

1, 15120, 3491888400, 1304290155168000, 601680868708529610000, 312696069714024464473125120, 175460887238127057573116837126400, 103865765423748548466734695459219968000, 63958974275578307119821712720619705931210000, 40596987692554701292235753375257230410967703200000

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 2.

S. Hassani, J-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 14.

Cf. A000442, A010050, A100734, A195394.

Cf. A381162, A381163, A381164, A381165.

a[n_]:=(10n)!/((n!)^3*(2n)!*(5n)!); Array[a,10,0]

G.f.: hypergeom([1/10, 3/10, 7/10, 9/10], [1, 1, 1], 2^8*5^5*x).

a(n) ~ 9*2^(3+8*n)*5^(1+5*n)/((1 + 24*n)*(1 + 60*n)*Pi^2).

A381165 a(n) = Sum_{k=0..n} binomial(2n,n)binomial(n, k)(5k)!/((k!)^3(2k)!).

Original entry on oeis.org

1, 122, 114126, 169305620, 307902541870, 628881704226972, 1384648756554128604, 3213280613371692112392, 7752574653184355259506670, 19272593072633780827550508620, 49062146831202726778631520779476, 127331178560917294198014376933764792, 335791906923524740189894975371277920796

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 128.

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 16.

Cf. A000442, A000984, A007318, A010050, A100734.

Cf. A381161, A381162, A381163, A381164.

a[n_]:=Sum[Binomial[2n,n]Binomial[n, k](5k)!/((k!)^3*(2k)!), {k, 0, n}]; Array[a, 13, 0]

G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1, 1, 1], 5^5*x/(1-4*x))/sqrt(1-4*x).
a(n) = binomial(2*n,n)*hypergeom([1/5, 2/5, 3/5, 4/5, -n], [1/2, 1, 1, 1], -5^5/4).
a(n) ~ 3^(n + 3/2) * 7^(n + 3/2) * 149^(n +3/2) / (4 * 5^7 * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025

A269947 Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 8, 9, 1, 0, 216, 251, 36, 1, 0, 13824, 16280, 2555, 100, 1, 0, 1728000, 2048824, 335655, 15055, 225, 1, 0, 373248000, 444273984, 74550304, 3587535, 63655, 441, 1, 0, 128024064000, 152759224512, 26015028256, 1305074809, 25421200, 214918, 784, 1

Offset: 0

Peter Luschny, Mar 22 2016

			Triangle starts:
1,
0, 1,
0, 1,       1,
0, 8,       9,       1,
0, 216,     251,     36,     1,
0, 13824,   16280,   2555,   100,   1,
0, 1728000, 2048824, 335655, 15055, 225, 1.

Peter Luschny, The P-transform.

Variant: A249677.
Cf. A007318 (order 0), A132393 (order 1), A269944 (order 2).
Cf. A000442, A000537, A255433.

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + (n-1)^3*T(n-1, k))) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

T[n_, k_] := T[n, k] = Which[n == k, 1, k < 0 || k > n, 0, True, T[n - 1, k - 1] + (n - 1)^3 T[n - 1, k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

T(n,1) = ((n-1)!)^3 for n>=1 (cf. A000442).
T(n,n-1) = (n*(n-1)/2)^2 for n>=1 (cf. A000537).
Row sums: Product_{k=1..n} ((k-1)^3+1) for n>=0 (cf. A255433).

A316862 Expansion of 1/(Sum_{k>=0} (k!)^3 x^k).

Original entry on oeis.org

1, -1, -7, -201, -13351, -1697705, -369575303, -127249900617, -65286578868455, -47651775381867241, -47688241963081263175, -63505249400026210723209, -109775495351620406817045415, -241236985075124408660287423529, -662075390371447206867029299628807

Offset: 0

Seiichi Manyama, Jul 15 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..181

1/(Sum_{k>=0} (k!)^b x^k): A167894 (b=1), A113871 (b=2), this sequence (b=3).

Cf. A000442.

a[n_] := -Sum[(k!)^3*a[n - k], {k, n}]; a[0] = 1; Array[a, 15, 0] (* Robert G. Wilson v, Jul 15 2018 *)
nmax = 20; CoefficientList[Series[1/Sum[k!^3 * x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 08 2020 *)

a(0) = 1, a(n) = -Sum_{k=1..n} (k!)^3 * a(n-k).

a(n) ~ -(n!)^3 * (1 - 2/n^3 - 13/n^6 - 39/n^7 - 78/n^8 - 518/n^9 - 3687/n^10 - ...). - Vaclav Kotesovec, Dec 08 2020

A351800 a(n) = [x^n] 1/Product_{j=1..n} (1 - j^3*x).

Original entry on oeis.org

1, 1, 73, 28800, 33120201, 83648533275, 393764054984212, 3103381708489548640, 37965284782803741391413, 681476650259874114533077575, 17184647574689079046814198039765, 588057239856779143071625300022102376, 26548105106818292578525347802793561068860

Offset: 0

Alois P. Heinz, Feb 19 2022

nonn

			a(2) = (1*1)^3 + (1*2)^3 + (2*2)^3 = 1 + 8 + 64 = 73.

Alois P. Heinz, Table of n, a(n) for n = 0..149

Cf. A000442, A001700, A007820, A088218, A098436, A269948, A298851, A351804.

b:= proc(n, k) option remember; `if`(k=0, 1,
      add(b(j, k-1)*j^3, j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..15);

Table[SeriesCoefficient[Product[1/(1 - k^3*x), {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 17 2025 *)

a(n) = Sum_{p in {1..n}^n : p_i <= p_{i+1}} Product_{j=1..n} p_j^3.
a(n) = A098436(2n-1,n-1) = A269948(2n,n).
a(n) ~ c * d^n * n^(3*n - 1/2), where d = 1.54371040458513693750053812318801418996889528987425... and c = 0.71526493063554190404119140313248864511356727815244... - Vaclav Kotesovec, May 13 2025

cp's OEIS Frontend

A225816 Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

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A134372 a(n) = ((2n)!)^2.

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PARI

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A134370 a(n) = ((2n+1)!)^(n+2).

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A134373 a(n) = ((2n)!)^3.

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A249677 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^3*x).

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A381161 a(n) = (10*n)!/((n!)^3*(2*n)!*(5*n)!).

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A381165 a(n) = Sum_{k=0..n} binomial(2*n,n)*binomial(n, k)*(5*k)!/((k!)^3*(2*k)!).

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A269947 Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.

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Maple

Mathematica

A381161 a(n) = (10n)!/((n!)^3(2n)!(5*n)!).

A381165 a(n) = Sum_{k=0..n} binomial(2n,n)binomial(n, k)(5k)!/((k!)^3(2k)!).