A036740 -id:A036740 - cp's OEIS frontend

A134371 a(n) = ((2n)!)^n.

1, 2, 576, 373248000, 2642908293365760000, 629238322197897601351680000000000, 12078744213598964456884373878200091017216000000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A134366, A134367, A134368, A134369, A134370, A134374, A134375.

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

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

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

Original entry on oeis.org

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).

A091868 a(n) = (n!)^(n+1).

Original entry on oeis.org

1, 1, 8, 1296, 7962624, 2985984000000, 100306130042880000000, 416336312719673760153600000000, 281633758444745849464726940024832000000000, 39594086612242519324387557078266845776303882240000000000

Offset: 0

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br), Mar 10 2004

Let f(x) be a monic polynomial of degree n. Let u be any number and let m be the matrix of values f(u+i-j) for i,j=1..n. Then the determinant of m is a(n). - T. D. Noe, Aug 23 2008
From Andrew Weimholt, Sep 23 2009: (Start)
Also, number of ways to assemble an n-simplex from n+1 labeled (n-1)-simplices with labeled vertices, where left-handed and right-handed counterparts are considered equivalent.
For n=2, we are constructing a triangle from 3 labeled line-segments with labeled endpoints. Solutions which differ by a rotation or a reflection are considered equivalent. Because reflections are equivalent, there is only 1 way to order the line-segments, and each line-segment can be oriented in 2 ways, so the total number of solutions is 2^3 = 8. For n=3, we are constructing a tetrahedron from 4 labeled triangles with labeled vertices. Without loss of generality, we can pick one labeled triangle to serve as our face of reference. For this face, we do not care which side of the triangle will face the interior of the tetrahedron as this just translates into a reflection of the tetrahedron, nor do we care about which rotation we pick as these just translate into rotations of the tetrahedron. From this reference triangle, there are 6 (=3!) ways to assign the remaining triangles to the faces of the tetrahedron, and each triangle can be oriented in 6 (=3!) ways (we can pick which side of the triangle will face the interior of the tetrahedron, and we can pick from 3 rotations). This gives 6^4 solutions.
Cf. A165644 (same idea, but reflections are distinct). A165642 and A165643 are the corresponding sequences for cubes instead of simplices. (End)
a(n) is the number of preference profiles in the stable marriage problem with n women and n men, where all the men rank women in the same order. Given such a profile, the Gale-Shapley men-proposing algorithm ends in n rounds. Equivalently, this is the number of preference profiles where all the women rank men in the same order. - Tanya Khovanova and MIT PRIMES STEP Senior group, May 23 2021
a(n-1) is the determinant of the n X n matrix with elements m(i,j) = s(n+i-1,j), 1 <= i <= n, 1 <= j <= n, where s(x,y) are the unsigned Stirling numbers of the first kind. - Fabio Visonà, May 22 2022

Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021.
Peter J. Taylor, Determinant of matrix with Stirling numbers as elements
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A036740.

[Factorial(n)^(n+1): n in [0..10]]; // Vincenzo Librandi, Nov 25 2015

(n!)^(n+1);
a[0]:=1:for n from 1 to 20 do a[n]:=product(n!, k=0..n) od: seq(a[n], n=0..8); # Zerinvary Lajos, Jun 11 2007
seq(mul(mul(j,j=1..n), k=0..n), n=0..8); # Zerinvary Lajos, Sep 21 2007

Table[(n!)^(n+1),{n,0,8}] (* Harvey P. Dale, Apr 30 2012 *)

a(n) = (n!)^(n+1) = a(n-1) * n^n * n!.
a(n) = A000178(n)*A002109(n), i.e., product of superfactorials and hyperfactorials. - Henry Bottomley, Nov 13 2009
a(n) ~ (2*Pi)^((n+1)/2) * n^((n+1)*(2*n+1)/2) / exp(n^2 + n - 1/12). - Vaclav Kotesovec, Jul 10 2015

Edited by N. J. A. Sloane, Oct 24 2009 at the suggestion of R. J. Mathar

a(9) from Vincenzo Librandi, Nov 25 2015

A086687 a(n) = binomial(n!,n).

Original entry on oeis.org

1, 1, 1, 20, 10626, 190578024, 189492294437160, 16321981448897949868560, 173117930575796585984663510371920, 300649979042426162195454856425990347600496000, 109109335363647915965076917082046041529603540914857910625920

Offset: 0

Jon Perry, Jul 28 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..30
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.

Cf. A036740.

Main diagonal of A180397.

Table[Binomial[n!,n],{n,0,12}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011*)

PARI
```
for(i=1,10,print1(","binomial(i!,i)))
```

From Vaclav Kotesovec, Jul 02 2016: (Start)
a(n) ~ (n!)^(n-1).
a(n) ~ 2^((n-1)/2) * Pi^((n-1)/2) * n^((2*n+1)*(n-1)/2) / exp(n^2-n-1/12).
(End)

a(0)=1 prepended by Alois P. Heinz, Nov 06 2016

A121997 Count up to n, n times.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7

Offset: 1

Franklin T. Adams-Watters, Sep 11 2006

The n-th block consists of n subblocks, each of which counts from 1 to n.
This a fractal sequence: removing the first instance of each value leaves the original sequence.
The first comment implies that this gives the column index of the n-th element of a sequence whose terms are coefficients, read by rows, of a sequence of matrices of size 1 X 1, 2 X 2, 3 X 3, etc.; cf. example. The row index is given by A238013(n), and the size of the matrix by A074279(n). - M. F. Hasler, Feb 16 2014

			Sequence begins:
  1;
  1,2;
  1,2;
  1,2,3;
  1,2,3;
  1,2,3;
  ...
The blocks of n subblocks of n terms (n=1,2,3,...) can be cast into a square matrices of order n; then the terms are equal to the index of the column they fall into.

Cf. A081489 (locations of new values), A075349 (locations of 1's).
Cf. A000290 (row lengths), A002411 (row sums), A036740 (row products).
Cf. A002024 and references there, esp. in PROG section.
Cf. A238013.

A121997(N=9)=concat(vector(N,i,concat(vector(i,j,vector(i,k,k))))) \\ Note: this creates a vector; use A121997()[n] to get the n-th term. - M. F. Hasler, Feb 16 2014

from sympy import integer_nthroot
def A121997(n): return 1+(n-(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k-1)*((k<<1)-1)//6-1)%k # Chai Wah Wu, Nov 04 2024

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)

A053986 a(n) = n^(n!).

Original entry on oeis.org

0, 1, 4, 729, 281474976710656, 752316384526264005099991383822237233803945956334136013765601092018187046051025390625

Offset: 0

Henry Bottomley, Apr 03 2000

Next term has 561 digits.

			a(3) = 729 because 3^3! = 3^6 = 729.

Cf. A036740, A050923, A100084.

Table[n^n!, {n, 0, 5}] (* Alonso del Arte, Jan 03 2011 *)

makelist(n^n!,n,0,6); /* Martin Ettl, Jan 13 2013 */

Sum_{n>=1} 1/a(n) = A100084. - Amiram Eldar, Nov 11 2020

One more term from Lior Manor, Nov 27 2001

A215910 a(n) = sum of the n-th power of the multinomial coefficients in row n of triangle A036038.

Original entry on oeis.org

1, 1, 5, 244, 354065, 25688403126, 141528428949437282, 83257152559805973052807833, 7012360438832401192319979008881500417, 109324223115831487504443410090345278639832867784010, 396327911646787133737309113762487915762995734538047874429637296650

Offset: 0

Paul D. Hanna, Aug 26 2012

nonn

			The sums of the n-th power of multinomial coefficients in row n of triangle A036038 begin:
a(1) = 1^1 = 1;
a(2) = 1^2 + 2^2 = 5;
a(3) = 1^3 + 3^3 + 6^3 = 244;
a(4) = 1^4 + 4^4 + 6^4 + 12^4 + 24^4 = 354065;
a(5) = 1^5 + 5^5 + 10^5 + 20^5 + 30^5 + 60^5 + 120^5 = 25688403126;
a(6) = 1^6 + 6^6 + 15^6 + 20^6 + 30^6 + 60^6 + 90^6 + 120^6 + 180^6 + 360^6 + 720^6 = 141528428949437282;
a(7) = 1^7 + 7^7 + 21^7 + 35^7 + 42^7 + 105^7 + 140^7 + 210^7 + 210^7 + 420^7 + 630^7 + 840^7 + 1260^7 + 2520^7 + 5040^7 = 83257152559805973052807833; ...
which also form a logarithmic generating function of an integer series:
L(x) = x + 5*x^2/2 + 244*x^3/3 + 354065*x^4/4 + 25688403126*x^5/5 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 84*x^3 + 88602*x^4 + 5137769389*x^5 +...+ A215911(n)*x^n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..30

Cf. A183610, A036038, A036740, A005651, A183240, A183235, A183236, A183237, A183238; A215911.

Cf. A326321.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i), k)/i!^k+b(n, i-1, k))
    end:
a:= n-> n!^n*b(n$3):
seq(a(n), n=0..12);  # Alois P. Heinz, Sep 11 2019

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, 1, b[n - i, Min[n - i, i], k]/i!^k + b[n, i - 1, k]];
a[n_] := n!^n b[n, n, n];
a /@ Range[0, 12] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

{a(n)=n!^n*polcoeff(1/prod(m=1, n, 1-x^m/m!^n +x*O(x^n)), n)}
for(n=1,15,print1(a(n),", "))

a(n) = [x^n/n!^n] * Product_{k=1..n} 1/(1 - x^k/k!^n) for n>=1, with a(0)=1.
Logarithmic derivative of A215911, ignoring the initial term a(0).
a(n) ~ (n!)^n = A036740(n). - Vaclav Kotesovec, Feb 19 2015
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Feb 19 2015

A291333 a(n) = [x^n] 1/(1 - x/(1 - 2^nx/(1 - 3^nx/(1 - 4^nx/(1 - 5^nx/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 5, 297, 485729, 38103228225, 220579355255364545, 134210828762693919568092033, 11583583466188874003924403353591815169, 183988806081826466732185672966967145613350641690625, 676960735217941793634104089611911809588055950029181968418342810625

Offset: 0

Ilya Gutkovskiy, Aug 22 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..30

Main diagonal of A290569.

Cf. A036740.

seq(coeff(series(numtheory:-cfrac([0,[1,1],seq([-i^n*x,1],i=1..n)]),x,n+1),x,n),n=0..15); # Robert Israel, Aug 22 2017

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-i^n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 10}]

a(n) = A290569(n,n).

a(n) ~ c * (n!)^n ~ c * 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12), where c = 1/QPochhammer(exp(-1)) = 1.9824409074128737036856824655613120156828827... - Vaclav Kotesovec, Aug 26 2017, updated Jul 21 2018

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

cp's OEIS Frontend

A134371 a(n) = ((2n)!)^n.

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Mathematica

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

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Maple

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A091868 a(n) = (n!)^(n+1).

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Magma

Maple

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A086687 a(n) = binomial(n!,n).

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PARI

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A121997 Count up to n, n times.

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PARI

Python

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

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Mathematica

PARI

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A053986 a(n) = n^(n!).

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Mathematica

Maxima

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A215910 a(n) = sum of the n-th power of the multinomial coefficients in row n of triangle A036038.

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A291333 a(n) = [x^n] 1/(1 - x/(1 - 2^nx/(1 - 3^nx/(1 - 4^nx/(1 - 5^nx/(1 - ...)))))), a continued fraction.