A079478 -id:A079478 - cp's OEIS frontend

A324403 a(n) = Product_{i=1..n, j=1..n} (i^2 + j^2).

1, 2, 400, 121680000, 281324160000000000, 15539794609114833408000000000000, 49933566483104048708063697937367040000000000000000, 19323883089768863178599626514889213871887405416448000000000000000000000000

Offset: 0

Vaclav Kotesovec, Feb 26 2019

nonn

Next term is too long to be included.

Cf. A079478, A324426, A324437, A324438, A324439, A324440, A367834.

Cf. A272244, A293290, A324402, A324443, A324425.

a:= n-> mul(mul(i^2+j^2, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^2+j^2, {i, 1, n}, {j, 1, n}], {n, 1, 10}]

a(n) = prod(i=1, n, prod(j=1, n, i^2+j^2)); \\ Michel Marcus, Feb 27 2019

from math import prod, factorial
def A324403(n): return (prod(i**2+j**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2<Chai Wah Wu, Nov 22 2023

a(n) ~ 2^(n*(n+1) - 3/4) * exp(Pi*n*(n+1)/2 - 3*n^2 + Pi/12) * n^(2*n^2 - 1/2) / (Pi^(1/4) * Gamma(3/4)).
a(n) = 2*n^2*a(n-1)*Product_{i=1..n-1} (n^2 + i^2)^2. - Chai Wah Wu, Feb 26 2019
For n>0, a(n)/a(n-1) = A272244(n)^2 / (2*n^6). - Vaclav Kotesovec, Dec 02 2023
a(n) = exp(2*Integral_{x=0..oo} (n^2/(x*exp(x)) - (cosh(n*x) - cos(n*x))/(x*exp((n + 1)*x)*(cosh(x) - cos(x)))) dx)/2^(n^2). - Velin Yanev, Jun 30 2025

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A009963 Triangle of numbers n!(n-1)!...(n-k+1)!/(1!2!...k!).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 24, 72, 24, 1, 1, 120, 1440, 1440, 120, 1, 1, 720, 43200, 172800, 43200, 720, 1, 1, 5040, 1814400, 36288000, 36288000, 1814400, 5040, 1, 1, 40320, 101606400, 12192768000, 60963840000, 12192768000, 101606400, 40320, 1

Offset: 0

N. J. A. Sloane

Product of all matrix elements of n X k matrix M(i,j) = i+j (i=1..n-k, j=1..k). - Peter Luschny, Nov 26 2012

These are the generalized binomial coefficients associated to the sequence A000178. - Tom Edgar, Feb 13 2014

			Rows start:
  1;
  1,   1;
  1,   2,    1;
  1,   6,    6,    1;
  1,  24,   72,   24,   1;
  1, 120, 1440, 1440, 120, 1;  etc.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000178, A007318, A060854, A090441.
Central column is A079478.
Columns include A010796, A010797, A010798, A010799, A010800.
Row sums give A193520.

A009963:= func< n,k | (1/Factorial(n+1))*(&*[ Factorial(n-j+1)/Factorial(j): j in [0..k]]) >;
[A009963(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 04 2022

(* First program *)
row[n_]:= Table[Product[i+j, {i,1,n-k}, {j,1,k}], {k,0,n}];
Array[row, 9, 0] // Flatten (* Jean-François Alcover, Jun 01 2019, after Peter Luschny *)
(* Second program *)
T[n_, k_]:= BarnesG[n+2]/(BarnesG[k+2]*BarnesG[n-k+2]);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 04 2022 *)

def A009963_row(n):
    return [mul(mul(i+j for j in (1..k)) for i in (1..n-k)) for k in (0..n)]
for n in (0..7): A009963_row(n)  # Peter Luschny, Nov 26 2012

def triangle_to_n_rows(n): #changing n will give you the triangle to row n.
    N=[[1]+n*[0]]
    for i in [1..n]:
        N.append([])
        for j in [0..n]:
            if i>=j:
                N[i].append(factorial(i-j)*binomial(i-1,j-1)*N[i-1][j-1]+factorial(j)*binomial(i-1,j)*N[i-1][j])
            else:
                N[i].append(0)
    return [[N[i][j] for j in [0..i]] for i in [0..n]]
    # Tom Edgar, Feb 13 2014

T(n,k) = T(n-1,k-1)*A008279(n,n-k) = A000178(n)/(A000178(k)*A000178(n-k)) i.e., a "supercombination" of "superfactorials". - Henry Bottomley, May 22 2002
Equals ConvOffsStoT transform of the factorials starting (1, 2, 6, 24, ...); e.g., ConvOffs transform of (1, 2, 6, 24) = (1, 24, 72, 24, 1). Note that A090441 = ConvOffsStoT transform of the factorials, A000142. - Gary W. Adamson, Apr 21 2008
Asymptotic: T(n,k) ~ exp((3/2)*k^2 - zeta'(-1) + 3/4 - (3/2)*n*k)*(1+n)^((1/2)*n^2 + n + 5/12)*(1+k)^(-(1/2)*k^2 - k - 5/12)*(1 + n - k)^(-(1/2)*n^2 + n*k - (1/2)*k^2 - n + k - 5/12)/(sqrt(2*Pi). - Peter Luschny, Nov 26 2012
T(n,k) = (n-k)!*C(n-1,k-1)*T(n-1,k-1) + k!*C(n-1,k)*T(n-1,k) where C(i,j) is given by A007318. - Tom Edgar, Feb 13 2014
T(n,k) = Product_{i=1..k} (n+1-i)!/i!. - Alois P. Heinz, Jun 07 2017
T(n,k) = BarnesG(n+2)/(BarnesG(k+2)*BarnesG(n-k+2)). - G. C. Greubel, Jan 04 2022

A039622 Number of n X n Young tableaux.

Original entry on oeis.org

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

A324437 a(n) = Product_{i=1..n, j=1..n} (i^4 + j^4).

Original entry on oeis.org

1, 2, 18496, 189567553208832, 53863903330477722171391434817536, 4194051697335929481600368256016484482740174637152337920000, 530545265060440849231458462212366841894726534233233018777709463062563850450708386692464640000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Cf. A079478, A324403, A324426, A324438, A324439, A324440, A367834.

Cf. A203677, A272247, A307215.

a:= n-> mul(mul(i^4+j^4, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^4 + j^4, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

from math import prod, factorial
def A324437(n): return (prod(i**4+j**4 for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**2)**2<Chai Wah Wu, Nov 26 2023

a(n) ~ c * 2^(n*(n+1)) * exp(Pi*n*(n+1)/sqrt(2) - 6*n^2) * (1 + sqrt(2))^(sqrt(2)*n*(n+1)) * n^(4*n^2 - 1), where c = A306620 = 0.23451584451404279281807143317500518660696293944961...

For n>0, a(n)/a(n-1) = A272247(n)^2 / (2*n^12). - Vaclav Kotesovec, Dec 01 2023

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A324426 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3).

Original entry on oeis.org

1, 2, 2592, 134425267200, 3120795915109442519040000, 180825857777547616919759624941086965760000000, 99356698720512072045648926659510730227553351200000695922065408000000000

Offset: 0

Vaclav Kotesovec, Feb 27 2019

nonn

Cf. A079478, A324403, A324437, A324438, A324439, A324440, A367834.

Cf. A272246, A307210, A367517, A367543.

a:= n-> mul(mul(i^3+j^3, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^3+j^3, {i, 1, n}, {j, 1, n}], {n, 1, 10}]

a(n) = prod(i=1, n, prod(j=1, n, i^3+j^3)); \\ Michel Marcus, Feb 27 2019

from math import prod, factorial
def A324426(n): return prod(i**3+j**3 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**3<Chai Wah Wu, Nov 26 2023

a(n) ~ A * 2^(2*n*(n+1) + 1/4) * exp(Pi*(n*(n+1) + 1/6)/sqrt(3) - 9*n^2/2 - 1/12) * n^(3*n^2 - 3/4) / (3^(5/6) * Pi^(1/6) * Gamma(2/3)^2), where A is the Glaisher-Kinkelin constant A074962.
a(n) = A079478(n) * A367543(n). - Vaclav Kotesovec, Nov 22 2023
For n>0, a(n)/a(n-1) = A272246(n)^2 / (2*n^9). - Vaclav Kotesovec, Dec 02 2023

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A367543 a(n) = Product_{i=1..n, j=1..n} (i^2 - i*j + j^2).

Original entry on oeis.org

1, 36, 777924, 51190934086656, 32435802373365731229926400, 483207398728525904876601066508152707481600, 350969035472356907726779584093506665415605824531908346799718400

Offset: 1

Vaclav Kotesovec, Nov 22 2023

nonn

Cf. A203312, A324403, A367542.

Cf. A324426, A079478.

Table[Product[Product[(i^2 - i*j + j^2), {i, 1, n}], {j, 1, n}], {n, 1, 10}]

from math import prod, factorial
def A367543(n): return (prod(i*(i-j)+j**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2 # Chai Wah Wu, Nov 22 2023

a(n) = A324426(n) / A079478(n).

a(n) ~ 3^(1/6) * Gamma(1/3)^2 * n^(2*n^2 - 1/3) / (2^(5/3) * Pi^(5/3) * exp(3*n^2 - (n^2 + n + 1/6)*Pi/sqrt(3))).

A306594 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i + j + k).

Original entry on oeis.org

1, 3, 144000, 455282248974336000000, 9608917807566747651759509633033255126040576000000000000

Offset: 0

Vaclav Kotesovec, Feb 27 2019

nonn

Next term is too long to be included.

Cf. A079478, A093884, A112332, A324425, A368722, A368723, A324441.

a:= n-> mul(mul(mul(i+j+k, i=1..n), j=1..n), k=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Jun 24 2023

Table[Product[i+j+k, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 6}]
Table[Product[k^(3*(n - k + 1) (n - k + 2)/2), {k, 1, n}] * Product[k^((3*n - k + 1) (3*n - k + 2)/2), {k, 1, 3*n}] / Product[k^(3*(2*n - k + 1) (2*n - k + 2)/2), {k, 1, 2*n}], {n, 1, 6}]
Clear[a]; a[n_] := a[n] = If[n == 1, 3, 3*n*a[n-1] * BarnesG[2+n]^3 * BarnesG[2+3*n]^3 * Gamma[1+2*n]^3 / (BarnesG[2+2*n]^6 * Gamma[1+3*n]^3)]; Table[a[n], {n, 1, 6}] (* Vaclav Kotesovec, Mar 28 2019 *)

a(n) = Product_{k=1..n} (BarnesG(k+2) * BarnesG(2*n+k+2) / BarnesG(n+k+2)^2).
a(n) = Product_{k=1..n} (k^(3*(n - k + 1)*(n - k + 2)/2)) * Product_{k=1..3*n} (k^((3*n - k + 1)*(3*n - k + 2)/2)) / Product_{k=1..2*n} (k^(3*(2*n - k + 1)*(2*n - k + 2)/2)).
a(n) ~ sqrt(Pi) * 3^(9*n^3/2 + 27*n^2/4 + 3*n + 3/8) * n^(n^3 + 3/8) / (A^(3/2) * 2^(4*n^3 + 9*n^2 + 6*n + 5/8) * exp(11*n^3/6 - Zeta(3)/(8*Pi^2) - 1/8)), where A is the Glaisher-Kinkelin constant A074962.

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A306760 a(n) = Product_{i=1..n, j=1..n} (i*j + 1).

Original entry on oeis.org

1, 2, 90, 705600, 4105057320000, 52487876090562232320000, 3487017405172854771910634342400000000, 2448893405298238642974553493547144534294528000000000000, 33257039167768610289435138215602132823918399655132218973388800000000000000000

Offset: 0

Vaclav Kotesovec, Mar 08 2019

nonn

Cf. A079478, A101686, A185141, A324444, A306765, A324589, A306907.

a:= n-> mul(mul(i*j+1, i=1..n), j=1..n):
seq(a(n), n=0..9);  # Alois P. Heinz, Jun 24 2023

Table[Product[i*j + 1, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
Table[n!^(2*n) * Product[Binomial[n + 1/j, n], {j, 1, n}], {n, 1, 10}]

a(n) ~ c * n^(n*(2*n+1) + 2*gamma) * (2*Pi)^n * exp(1/6 + log(n)^2 - 2*n^2), where c = 1/A306765 and gamma is the Euler-Mascheroni constant A001620.

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A324425 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^2 + j^2 + k^2).

Original entry on oeis.org

1, 3, 5668704, 550388591715704109656479285248, 152455602303300418998634460043817052571893573096619261814850281699755319515987050496

Offset: 0

Vaclav Kotesovec, Feb 27 2019

nonn

(a(n)^(1/n^3))/n^2 tends to 0.828859579669279... = A306617.

Cf. A079478, A306594, A368722, A368723.

Cf. A324403, A306617, A368622, A368623.

a:= n-> mul(mul(mul(i^2+j^2+k^2, i=1..n), j=1..n), k=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^2+j^2+k^2, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 6}]
Clear[a]; a[n_] := a[n] = If[n == 1, 3, a[n-1] * Product[k^2 + j^2 + n^2, {j, 1, n}, {k, 1, n}]^3 * (3*n^2) / (Product[k^2 + 2*n^2, {k, 1, n}]^3)]; Table[a[n], {n, 1, 6}] (* Vaclav Kotesovec, Mar 27 2019 *)

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A324438 a(n) = Product_{i=1..n, j=1..n} (i^5 + j^5).

Original entry on oeis.org

1, 2, 139392, 305013568273920000, 1174837791623127613548781790822400000000, 139642003782073074626249921818187528362524804267528306032640000000000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Cf. A079478, A324403, A324426, A324437, A324439, A324440, A367834.

Cf. A272248, A367679.

a:= n-> mul(mul(i^5 + j^5, i=1..n), j=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Nov 26 2023

Table[Product[i^5 + j^5, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

from math import prod, factorial
def A324438(n): return prod(i**5+j**5 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**5<Chai Wah Wu, Nov 26 2023

a(n) ~ c * 2^(2*n*(n+1)) * phi^(sqrt(5)*n*(n+1)) * exp(Pi*sqrt(phi)*n*(n+1)/5^(1/4) - 15*n^2/2) * n^(5*n^2 - 5/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 0.1574073828647726237455544898360432469056972905505624900871695...
a(n) = A367679(n) * A079478(n). - Vaclav Kotesovec, Nov 26 2023
For n>0, a(n)/a(n-1) = A272248(n)^2 / (2*n^15). - Vaclav Kotesovec, Dec 02 2023

a(0)=1 prepended by Alois P. Heinz, Nov 26 2023

cp's OEIS Frontend

A324403 a(n) = Product_{i=1..n, j=1..n} (i^2 + j^2).

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A009963 Triangle of numbers n!(n-1)!...(n-k+1)!/(1!2!...k!).

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A039622 Number of n X n Young tableaux.

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A324437 a(n) = Product_{i=1..n, j=1..n} (i^4 + j^4).

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A324426 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3).

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A367543 a(n) = Product_{i=1..n, j=1..n} (i^2 - i*j + j^2).

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A306594 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i + j + k).

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