A324403 -id:A324403 - cp's OEIS frontend

A324440 a(n) = Product_{i=1..n, j=1..n} (i^7 + j^7).

1, 2, 8520192, 956147263254051187507200, 790929096572487518050439299107158612099228070051840000, 266108022587896795750359251172229660295854509829286134803404773931312693787460334360985600000000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

For m>1, Product_{j=1..n, k=1..n} (j^m + k^m) ~ c(m) * exp(n*(n+1)*s(m) - m*n*(n-2)/2) * n^(m*(n^2 - 1/4)), where s(m) = Sum_{j>=1} (-1)^(j+1)/(j*(1 + m*j)) and c(m) is a constant (dependent only on m). Equivalently, s(m) = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/m). - Vaclav Kotesovec, Dec 01 2023

Cf. A079478, A324403, A324426, A324437, A324438, A324439, A367834.

a:= n-> mul(mul(i^7 + j^7, i=1..n), j=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Nov 26 2023

Table[Product[i^7+j^7, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

from math import prod, factorial
def A324440(n): return prod(i**7+j**7 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**7<Chai Wah Wu, Nov 26 2023

Limit_{n->oo} (a(n)^(1/n^2))/n^7 = 2^(3/2) * (cos(3*Pi/14) / tan(Pi/7))^sin(3*Pi/14) / ((cos(Pi/14)*tan(3*Pi/14))^sin(Pi/14) * (sin(Pi/7)*tan(Pi/14))^cos(Pi/7)) * exp((Pi/sin(Pi/7) - 21)/2) = 0.0334234967249533921390751418772468470887965377...
From Vaclav Kotesovec, Dec 01 2023: (Start)
a(n) ~ c * exp(n*(n+1)*s - 7*n*(n-2)/2) * n^(7*(n^2 - 1/4)), where
s = Sum_{j>=1} (-1)^(j+1)/(j*(1 + 7*j)) = Pi/(2*sin(Pi/7)) + 3*log(2)/2 - 7 - cos(Pi/7) * log(2*sin(Pi/14)^2) - log(2*sin(3*Pi/14)^2) * sin(Pi/14) + log(cos(3*Pi/14)*cos(Pi/7) / sin(Pi/7)) * sin(3*Pi/14) = 0.10150386842315637912206687298894641634315636548242136512503... and
c = 0.068056503846689328929612652207251071282623125565150941566636264805878144...
Equivalently, s = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/7). (End)

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

A324443 a(n) = Product_{i=1..n, j=1..n} (1 + i^2 + j^2).

Original entry on oeis.org

1, 3, 972, 437987088, 1396064690700615936, 100943980553724942717460016640000, 408685260379151918936869901376463191556211834880000, 193581283410907012468703321819613695893448022144552623141894180044800000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Product_{i>=1, j>=1} (1 + 1/(i^2 + j^2)) is divergent.

Cf. A101686, A324403, A324425, A324444, A306398.

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

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

a(n) ~ c * 2^(n*(n+1)) * exp(Pi*n*(n+1)/2 - 3*n^2) * n^(2*n^2 + (Pi - 1)/2), where c = A306398 = 0.1740394919107672354475619059102344818913844938434521480869...

a(n) / A324403(n) ~ d * n^(Pi/2), where d = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313...

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

A306729 a(n) = Product_{i=0..n, j=0..n} (i! + j!).

Original entry on oeis.org

2, 16, 5184, 9559130112, 109045776752640000000000, 27488263744928988967331390258832998400000000000, 1147897050240877062218236820013018349788772091106840426434074807527014400000000000000

Offset: 0

Vaclav Kotesovec, Mar 06 2019

nonn

Cf. A000178, A055462, A079478, A203482, A217757, A323717, A324403, A325052.

Table[Product[i! + j!, {i, 0, n}, {j, 0, n}], {n, 0, 7}]
Clear[a]; a[n_] := a[n] = If[n == 0, 2, a[n-1] * Product[k! + n!, {k, 0, n}]^2 / (2*n!)]; Table[a[n], {n, 0, 7}] (* Vaclav Kotesovec, Mar 27 2019 *)
Table[Product[Product[k! + j!, {k, 0, j}], {j, 1, n}]^2 / (2^(n-1) * BarnesG[n + 2]), {n, 0, 7}] (* Vaclav Kotesovec, Mar 27 2019 *)

from math import prod, factorial as f
def a(n): return prod(f(i)+f(j) for i in range(n) for j in range(n))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Feb 16 2021

a(n) ~ c * 2^(n^2/2 + 2*n) * Pi^(n^2/2 + n) * n^(2*n^3/3 + 2*n^2 + 11*n/6 + 5/2) / exp(8*n^3/9 + 2*n^2 + n), where c = A324569 = 62.14398692334529025548974541735...

a(n) = a(n-1) * A323717(n)^2 / (2*n!). - Vaclav Kotesovec, Mar 28 2019

A324402 a(n) = Product_{i=1..n, j=1..n} (2*i + j).

Original entry on oeis.org

1, 3, 360, 6350400, 36212520960000, 117563342374788710400000, 337905477880065368190647009280000000, 1234818479230749311108497004714406224855040000000000, 7795494015765035913020359514023640290443493305037073940480000000000000

Offset: 0

Vaclav Kotesovec, Feb 26 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..25

Cf. A059486, A079478, A324403, A367956, A367957, A367958.

f:= n -> mul((2*i+n)!/(2*i)!,i=1..n):
map(f, [$0..10]); # Robert Israel, Feb 27 2019

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

a(n) ~ sqrt(A/Pi) * 3^(9*n*(n+1)/4 + 11/24) * n^(n^2 - 11/24) / (2^(n^2 + 3*n/2 + 17/24) * exp(3*n^2/2 + 1/24)), where A is the Glaisher-Kinkelin constant A074962.
a(n) = 3*n*a(n-1)*Product_{i=1..n-1} (2*i+n)(2*n+i). - Chai Wah Wu, Feb 26 2019
a(n) = a(n-1) * (3*n)! * (3*n-2)!!/((2*n)! * n!!). - Robert Israel, Feb 27 2019

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

A293290 a(n) = Product_{1 <= j <= k <= n} (k^2 + j^2).

Original entry on oeis.org

1, 2, 80, 187200, 50918400000, 2675955409920000000, 40702283662588674048000000000, 250658664786823821917343252480000000000000, 832906513114759565863066815448211678822400000000000000000, 1919381816160714520414106848157314737202346840876384256000000000000000000000

Offset: 0

Velin Yanev, Oct 05 2017

Suggested by Omar E. Pol from A264596 formula.

Cf. A101686, A203475, A324403.

Table[Product[k^2 + j^2, {k, 1, n}, {j, 1, k}], {n, 0, 10}]

[prod([prod([k^2+j^2 for j in range(1,k+1)]) for k in range(1,n+1)]) for n in range(10)] # Danny Rorabaugh, Oct 16 2017

a(n) ~ sqrt(Gamma(1/4)) * Pi^(-1/8) * 2^(n^2/2 + n - 1/8) * exp(Pi*n*(n+1)/4 - 3*n^2/2 - n + Pi/24) * n^(n*(n+1) + 1/4). - Vaclav Kotesovec, Feb 26 2019

A368065 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5ij + j^2).

Original entry on oeis.org

1, 7, 44100, 3210672937500, 12804360424787610000000000, 8591751256288909159255104643281750000000000, 2333034616280404811605303958158227652934766912996000000000000000000

Offset: 0

Vaclav Kotesovec, Dec 10 2023

nonn

Cf. A367543, A324403, A367542, A079478^2, A368067, A368064, A368066.

Cf. A368068, A368069.

Table[Product[i^2 + 5*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

a(n) ~ c * 7^(7*n*(n+1)/2) * ((5-sqrt(21))/2)^(sqrt(21)*n*(n+1)/2) * n^(2*n^2 - 4/3) / exp(3*n^2), where c = A368069.

A368622 a(n) = Product_{j=1..n, k=1..n} (j^2 + k^2 + n^2).

Original entry on oeis.org

1, 3, 5832, 172907569296, 419358815743567702818816, 267800543010963952830647446563000000000000, 110831581527076064529150462985910455129725821244148698662830080000

Offset: 0

Vaclav Kotesovec, Jan 01 2024

nonn

The limit has a closed form. In Mathematica: Exp[Integrate[Log[x^2 + y^2 + 1], {x,0,1}, {y,0,1}]]. The output is extremely large.

Cf. A272244, A324403, A324425, A368720, A368721.

Table[Product[j^2 + k^2 + n^2, {j, 1, n}, {k, 1, n}], {n, 0, 10}]

Limit_{n->oo} a(n)^(1/(n^2)) / n^2 = exp(Integral_{x=0..1, y=0..1} log(x^2 + y^2 + 1) dy dx) = 1.6143980185761253961882683158432481977126507900460725431661...

A203511 a(n) = Product_{1 <= i < j <= n} (t(i) + t(j)); t = A000217 = triangular numbers.

Original entry on oeis.org

1, 1, 4, 252, 576576, 87178291200, 1386980110791475200, 3394352757964564324299571200, 1760578659300452732262852600316664217600, 255323290537547288382098619855584488593426606981120000

Offset: 0

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203512.

See A093883 for a guide to related sequences.

Cf. A000217, A203512, A293290, A324403, A324443.

t:= n-> n*(n+1)/2:
a:= n-> mul(mul(t(i)+t(j), i=1..j-1), j=2..n):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 23 2017

f[j_] := j (j + 1)/2; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]               (* A203511 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A203512 *)
Table[Product[k*(k+1)/2 + j*(j+1)/2, {k, 1, n}, {j, 1, k-1}], {n, 0, 10}] (* Vaclav Kotesovec, Sep 07 2023 *)

a(n) ~ c * 2^n * exp(n^2*(Pi/4 - 3/2) + n*(Pi/2 + 1)) * n^(n^2 - n - 2 - Pi/8), where c = 0.2807609661547466473998991675307759198889389396430915721129636653... - Vaclav Kotesovec, Sep 07 2023

Name edited by Alois P. Heinz, Jul 23 2017

a(0)=1 prepended by Alois P. Heinz, Jul 29 2017

A367944 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*j^2).

Original entry on oeis.org

1, 6, 27216, 1344924798336, 3605580335899213007486976, 1648055031941075082958467426002632704000000, 312704667066499295437237787452750428210311485710262201221120000000

Offset: 0

Vaclav Kotesovec, Dec 05 2023

nonn

In general, for d>0, Product_{i=1..n, j=1..n} (i^2 + d*j^2) ~ c(d) * n^(2*n^2 - 1/2) * (d+1)^(n*(n+1)) * d^(-n/2) * exp(n*(n+1)*(Pi*d/2 - (d-1)*arctan(sqrt(d))) / sqrt(d) - 3*n^2), where c(d) is a constant (dependent only on d).

c(1) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4), cf. A324403.

Cf. A324403 (d=1), A367941 (d=2), A367942 (d=3), A367943 (d=4).

Cf. A367542, A367543, A367958.

Table[Product[i^2+5*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

a(n) ~ c * n^(2*n^2 - 1/2) * 6^(n*(n+1)) * 5^(-n/2) * exp(n*(n+1)*(5*Pi/2 - 4*arctan(sqrt(5)))/sqrt(5) - 3*n^2), where c = 0.4431081869167792949266065295798218232844989957987096447783995373751372668...

A368066 a(n) = Product_{i=1..n, j=1..n} (i^2 + 6ij + j^2).

Original entry on oeis.org

1, 8, 73984, 10027173445632, 93867986947606492024406016, 185865459466664040069739311383413462872883200, 186896871826703385639703785281909582209471190408233074664996759142400

Offset: 0

Vaclav Kotesovec, Dec 10 2023

nonn

In general, for d >= -1, Product_{i=1..n, j=1..n} (i^2 + d*i*j + j^2) ~ c(d) * (d+2)^((d+2)*n*(n+1)/2) * n^(2*n^2 - 1/2 - d/6) / ((d/2 + sqrt(d^2/4 - 1))^(sqrt(d^2 - 4)*n*(n+1)/2) * exp(3*n^2)), where c(d) is a constant (dependent only on d).
c(-1) = 3^(1/6) * exp(Pi/(6*sqrt(3))) * Gamma(1/3)^2 / (2*Pi)^(5/3).
c(0) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4).
c(1) = 3^(5/12) * exp(Pi/(12*sqrt(3))) * Gamma(1/3) / (2*Pi)^(4/3).
c(2) = A^2 / (2^(1/6) * exp(1/6) * Pi), where A = A074962.
c(3) = 2^((sqrt(5) - 9)/6) * sqrt(5) * (1 + sqrt(5))^(1/2 - sqrt(5)/6) / Pi.
c(4) = 2^((sqrt(3) - 1)/6) * 3^(13/24) * (1 + sqrt(3))^(1/2 - 1/sqrt(3)) / (Pi^(7/12) * Gamma(1/4)^(1/3) * Gamma(1/3)^(1/2)).
c(5) = A368069.
c(6) = 2^(25/8) * (1 + sqrt(2))^(3/4 - 2*sqrt(2)/3) / (Pi^(1/4) * Gamma(1/8) * Gamma(1/4)^(1/2)).
Special (non-integer) case: Product_{i=1..n, j=1..n} (i^2 + (d + 1/d)*i*j + j^2) ~ A^(2/d) * (Product_{j=1..d} Gamma(j/d)^(2*j/d)) * (d+1)^((d/2 + 1 + 1/(2*d))*2*n*(n+1) + (d+1)^2/(6*d) + 1/6) * n^(2*n^2 - d/6 - 1/2 - 1/(6*d)) / ((2*Pi)^((d+1)/2) * exp(3*n^2 + 1/(6*d)) * d^((d+1)*n*(n+1) - 1/(6*d))), where A = A074962 is the Glaisher-Kinkelin constant.

Cf. A367543 (d=-1), A324403 (d=0), A367542 (d=1), A079478^2 (d=2), A368067 (d=3), A368064 (d=4), A368065 (d=5).

Cf. A368069, A368068.

Table[Product[i^2 + 6*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

a(n) ~ 2^(12*n*(n+1) + 25/8) * n^(2*n^2 - 3/2) / (Pi^(1/4) * Gamma(1/4)^(1/2) * Gamma(1/8) * (1 + sqrt(2))^(2*sqrt(2)*(6*n*(n+1) + 1)/3 - 3/4) * exp(3*n^2)).

cp's OEIS Frontend

A324440 a(n) = Product_{i=1..n, j=1..n} (i^7 + j^7).

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A324443 a(n) = Product_{i=1..n, j=1..n} (1 + i^2 + j^2).

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A306729 a(n) = Product_{i=0..n, j=0..n} (i! + j!).

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A324402 a(n) = Product_{i=1..n, j=1..n} (2*i + j).

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A293290 a(n) = Product_{1 <= j <= k <= n} (k^2 + j^2).

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A368065 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*i*j + j^2).

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A368622 a(n) = Product_{j=1..n, k=1..n} (j^2 + k^2 + n^2).

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A203511 a(n) = Product_{1 <= i < j <= n} (t(i) + t(j)); t = A000217 = triangular numbers.

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A368065 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5ij + j^2).

A368066 a(n) = Product_{i=1..n, j=1..n} (i^2 + 6ij + j^2).