A107254 -id:A107254 - cp's OEIS frontend

A039622 Number of n X n Young tableaux.

1, 1, 2, 42, 24024, 701149020, 1671643033734960, 475073684264389879228560, 22081374992701950398847674830857600, 220381378415074546123953914908618547085974856000, 599868742615440724911356453304513631101279740967209774643120000

Offset: 0

Floor van Lamoen

Number of arrangements of 1,2,...,n^2 in an n X n array such that each row and each column is increasing. The problem for a 5 X 5 array was recently posed and solved in the College Mathematics Journal. See the links.
This is the factor g_n that appears in a conjectured formula for 2n-th moment of the Riemann zeta function on the critical line. (See Conrey articles.) - Michael Somos, Apr 15 2003 [Comment revised by N. J. A. Sloane, Jun 21 2016]
Number of linear extensions of the n X n lattice. - Mitch Harris, Dec 27 2005

			Using the hook length formula, a(4) = (16)!/(7*6^2*5^3*4^4*3^3*2^2) = 24024.

M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 284.

Alois P. Heinz, Table of n, a(n) for n = 0..30
P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
Joerg Arndt, The a(3)=42 3 X 3 Young tableaux
J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 349.
J. B. Conrey, Review of H. Iwaniec, "Lectures on the Riemann Zeta Function" (AMS, 2014), Bull. Amer. Math. Soc., 53 (No. 3, 2016), 507-512.
P.-O. Dehaye, Combinatorics of the lower order terms in the moment conjectures: the Riemann zeta function, arXiv preprint arXiv:1201.4478 [math.NT], 2012.
J. S. Frame, G. de B. Robinson and R. M. Thrall, The hook graphs of a symmetric group, Canad. J. Math. 6 (1954), pp. 316-324.
Curtis Greene and Brady Haran, Shapes and Hook Numbers, Numberphile video (2016)
Curtis Greene and Brady Haran, Shapes and Hook Numbers (extra footage) (2016)
Zachary Hamaker and Eric Marberg, Atoms for signed permutations, arXiv:1802.09805 [math.CO], 2018.
Alejandro H. Morales, I. Pak, and G. Panova, Why is pi < 2 phi?, Preprint, 2016; The American Mathematical Monthly, Volume 125, 2018 - Issue 8.
Alan H. Rapoport (proposer), Solution to Problem 639: A Square Young Tableau, College Mathematics Journal, Vol. 30 (1999), no. 5, pp. 410-411.
Index entries for sequences related to Young tableaux.

Main diagonal of A060854.
Also a(2)=A000108(2), a(3)=A005789(3), a(4)=A005790(4), a(5)=A005791(5).
Cf. A000178, A000984, A079478, A088020, A107252, A107254, A127223, A153452.

A039622:= func< n | n eq 0 select 1 else Factorial(n^2)*(&*[Factorial(j)/Factorial(n+j): j in [0..n-1]]) >;
[A039622(n): n in [0..12]]; // G. C. Greubel, Apr 21 2021

a:= n-> (n^2)! *mul(k!/(n+k)!, k=0..n-1):
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 10 2012

a[n_]:= (n^2)!*Product[ k!/(n+k)!, {k, 0, n-1}]; Table[ a[n], {n, 0, 12}] (* Jean-François Alcover, Dec 06 2011, after Pari *)

PARI
```
a(n)=(n^2)!*prod(k=0,n-1,k!/(n+k)!)
```

def A039622(n): return factorial(n^2)*product( factorial(j)/factorial(n+j) for j in (0..n-1))
[A039622(n) for n in (0..12)] # G. C. Greubel, Apr 21 2021

a(n) = (n^2)! / Product_{k=1..2n-1} k^(n - |n-k|).
a(n) = 0!*1!*...*(k-1)! *(k*n)! / ( n!*(n+1)!*...*(n+k-1)! ) for k=n.
a(n) = A088020(n)/A107254(n) = A088020(n)*A000984(n)/A079478(n). - Henry Bottomley, May 14 2005
a(n) = A153452(prime(n)^n). - Naohiro Nomoto, Jan 01 2009
a(n) ~ sqrt(Pi) * n^(n^2+11/12) * exp(n^2/2+1/12) / (A * 2^(2*n^2-7/12)), where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 10 2015
From Peter Luschny, May 20 2019: (Start)
a(n) = (G(1+n)*G(2+n)^(2-n)*(n^2)!*(G(3+n)/Gamma(2+n))^(n-1))/(G(1+2*n)*n!) where G(x) is the Barnes G function.
a(n) = A127223(n) / A107252(n). (End)
a(n) = (Gamma(n^2 +1)/Gamma(n+1))*(G(n+1)*G(n+2)/G(2*n+1)), where G(n) is the Barnes G-function. - G. C. Greubel, Apr 21 2021
a(n+2) = (n+2) * A060856(n+1) for n >= 0. - Tom Copeland, May 30 2022

A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.

Original entry on oeis.org

1, 1, 6, 1440, 36288000, 184354652160000, 309071606732292096000000, 254046582743105184577722777600000000, 142518177743863255019484504453155074867200000000000

Offset: 0

Henry Bottomley, May 14 2005

nonn

			a(3) = 1!*2!*3!*4!*5!/(1!*2!*3!*1!*2!) = 34560/24 = 1440.

Cf. A000178, A001700, A005130, A009963, A110131, A112332, A107254, A000142.

[1] cat [(&*[Factorial(n+k)/Factorial(k+1): k in [0..n-1]]): n in [1..10]]; // G. C. Greubel, May 21 2019

Table[Product[(n+k)!/(k+1)!,{k,0,n-1}],{n,0,10}] (* Alexander Adamchuk, Jul 10 2006 *)
a[n_] := (BarnesG[1 + 2 n] n! ((BarnesG[2 + n] Gamma[2 + n])/ BarnesG[3 + n])^(-1 + n))/BarnesG[2 + n]^2; Table[a[n], {n, 0, 10}] (* Peter Luschny, May 20 2019 *)

{a(n) = prod(k=0,n-1, (n+k)!/(k+1)!)}; \\ G. C. Greubel, May 21 2019

[product(factorial(n+k)/factorial(k+1) for k in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 21 2019

a(n) = (n+1)!*(n+2)!*...*(2n-1)!/(1!*2!*...*(n-1)!).
a(n) = A000178(2n-1)/(A000178(n)*A000178(n-1)).
a(n) = A079478(n)/A001813(n).
a(n) = A079478(n-1)*A006963(n+1).
a(n) = A107251(n)/A000108(n).
a(n) = A107251(n-1)*A009445(n-1).
a(n) = A107254(n)/A000142(n).
a(n) = A009963(2n-1, n-1).
a(n) = A009963(2n-1, n).
a(n) = (G(1+2*n)*n!*((G(2+n)*Gamma(2+n))/G(3+n))^(n-1))/G(2+n)^2, where G(x) is the Barnes G function. - Peter Luschny, May 20 2019
a(n) ~ A * 2^(2*n^2 - 7/12) * n^(n^2 - n - 5/12) / (sqrt(Pi) * exp(3*n^2/2 - n + 1/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 21 2019

A136411 a(n) = Product_{k=1..n} (2k-1)^(2n-2*k+1).

Original entry on oeis.org

1, 3, 135, 212625, 21097715625, 207248662456171875, 291128066470548703880859375, 79746389028864195813528714933837890625, 5570294521107277357810167397301815834831695556640625

Offset: 1

Ctibor O. Zizka, Mar 31 2008

G. C. Greubel, Table of n, a(n) for n = 1..29

Cf. A136807, A087315, A000178, A055209.

[(&*[(2*k-1)^(2*n-2*k+1): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[(2*k - 1)^(2*n - 2*k + 1), {k, 1, n}], {n, 1, 10}] (* Stefan Steinerberger, May 18 2008 *)
sf[n_] := BarnesG[n + 2]; a[n_] := sf[2 n - 1]/(2^(n (n - 1)) sf[n - 1]^2); Table[a[n], {n, 1, 10}]  (* Robert Coquereaux, Apr 02 2013 *)

a(n) = prod(k=1, n, (2*k-1)^(2*n-2*k+1)) \\ Anders Hellström, Sep 16 2015

a(n) = A107254(n) / 2^(n*(n - 1)).
a(n) = sf(2*n-1) / (2^(n*(n-1)) * sf(n-1)^2), n >= 1, where sf(n) = BarnesG(n + 2) is the superfactorial defined in A000178. - Robert Coquereaux, Apr 02 2013
a(n) ~ A * 2^(n^2 + n - 1/12) * n^(n^2 + 1/12) / exp(3*n^2/2 + 1/12), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

More terms from Stefan Steinerberger, May 18 2008

cp's OEIS Frontend

A039622 Number of n X n Young tableaux.

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A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.

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A136411 a(n) = Product_{k=1..n} (2*k-1)^(2*n-2*k+1).

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A325053 a(n) = Product_{i=0..n, j=0..n} (i! + j! + 1).

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A136411 a(n) = Product_{k=1..n} (2k-1)^(2n-2*k+1).