cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

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

Original entry on oeis.org

1, 2, 400, 121680000, 281324160000000000, 15539794609114833408000000000000, 49933566483104048708063697937367040000000000000000, 19323883089768863178599626514889213871887405416448000000000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 26 2019

Keywords

Comments

Next term is too long to be included.

Crossrefs

Cf. A079478, A324426, A324437, A324438, A324439, A324440, A367834.
Cf. A272244, A293290, A324402, A324443, A324425.

Programs

  • Maple
    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
  • Mathematica
    Table[Product[i^2+j^2, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
  • PARI
    a(n) = prod(i=1, n, prod(j=1, n, i^2+j^2)); \\ Michel Marcus, Feb 27 2019
  • Python
    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

Formula

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

Extensions

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

A323540 a(n) = Product_{k=0..n} (k^2 + (n-k)^2).

Original entry on oeis.org

0, 1, 32, 2025, 204800, 30525625, 6307891200, 1727713080625, 606076928000000, 265058191985900625, 141409376995328000000, 90403125002859606705625, 68229510086445571768320000, 60026603304487418050791015625, 60893916244529680380723200000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Links

Crossrefs

Cf. A272244, A323425, A323541, A323542, A323543, A323544, A323545, A323546.
Cf. 2*A000330 and A006331 (with sum instead of product).

Programs

  • Magma
    m:=2; [(&*[k^m + (n-k)^m: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jan 18 2019
  • Mathematica
    Table[Product[k^2+(n-k)^2, {k, 0, n}], {n, 0, 20}]
  • PARI
    m=2; vector(20, n, n--; prod(k=0,n, k^m + (n-k)^m)) \\ G. C. Greubel, Jan 18 2019
  • Sage
    m=2; [product(k^m +(n-k)^m for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jan 18 2019

Formula

a(n) ~ exp((Pi-4)*n/2) * n^(2*n+2).

A203475 a(n) = Product_{1 <= i < j <= n} (i^2 + j^2).

Original entry on oeis.org

1, 5, 650, 5525000, 5807194900000, 1226800120038480000000, 77092420109247492627600000000000, 2001314057760220784660590245696000000000000000, 28468550112906756205383102673584071297339520000000000000000000
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2012

Keywords

Comments

Each term divides its successor, as in A203476.

Links

Crossrefs

Cf. A000290, A093883, A202768, A203476, A272244, A293290, A323755, A324403, A367517.

Programs

  • Magma
    [(&*[(&*[j^2 + k^2: k in [1..j]])/(2*j^2): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023
  • Maple
    a:= n-> mul(mul(i^2+j^2, i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017
  • Mathematica
    f[j_]:= j^2; z = 15;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]           (* A203475 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A203476 *)
  • SageMath
    [product(product(j^2+k^2 for k in range(1,j)) for j in range(1,n+1)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023

Formula

a(n) ~ c * 2^(n^2/2) * exp(Pi*n*(n+1)/4 - 3*n^2/2 + n) * n^(n*(n-1) - 3/4), where c = A323755 = sqrt(Gamma(1/4)) * exp(Pi/24) / (2*Pi)^(9/8) = 0.274528350333552903800408993482507428142383783773190451181... - Vaclav Kotesovec, Jan 26 2019

Extensions

Name edited by Alois P. Heinz, Jul 23 2017

A272246 a(n) = Product_{k=0..n} (n^3 + k^3).

Original entry on oeis.org

0, 2, 1152, 1428840, 3488808960, 15044494500000, 105235903511101440, 1119277024472896248960, 17216259547948971039129600, 368066786222106315186876633600, 10591209807103301277597696000000000, 399472472359100444604916002033020774400
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 23 2016

Keywords

Crossrefs

Cf. A126804, A272244, A272247, A272248, A367823, A367833.
Cf. A255433, A323541, A324426.

Programs

  • Mathematica
    Table[Product[n^3+k^3,{k,0,n}],{n,0,12}]

Formula

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

A272247 a(n) = Product_{k=0..n} (n^4 + k^4).

Original entry on oeis.org

0, 2, 8704, 104372388, 3087748038656, 194985808028125000, 23467500289618753093632, 4938279594477505466022892304, 1699491802948673617630927695904768, 907214902722906584628050661111614570016, 719684491044699824651608981274624000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 23 2016

Keywords

Crossrefs

Cf. A126804, A272244, A272246, A272248, A367823, A367833.
Cf. A255434, A323542.

Programs

  • Mathematica
    Table[Product[n^4+k^4,{k,0,n}],{n,0,10}]

Formula

a(n) ~ 2^(n+1/2) * (1+sqrt(2))^(sqrt(2)*n) * n^(4*n + 4) / exp((4 - Pi/sqrt(2))*n).

A272248 a(n) = Product_{k=0..n} (n^5 + k^5).

Original entry on oeis.org

0, 2, 67584, 7924375800, 2876035930521600, 2693451205324687500000, 5648896640332217707816550400, 23819277009290664033936067933468800, 185754160490281505676324140907107450880000, 2507604631016507710974687639612411760216253760000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 23 2016

Keywords

Comments

In general, for p>=1, Product_{k=0..n} (n^p + k^p) ~ sqrt(2) * n^(p*(n+1)) * exp(n*Sum_{j>=1} (-1)^(j+1) / (j*(1 + j*p))).

Crossrefs

Cf. A126804, A272244, A272246, A272247, A367823, A367833.
Cf. A255435, A323543, A324438.

Programs

  • Mathematica
    Table[Product[n^5+k^5,{k,0,n}],{n,0,10}]

Formula

a(n) ~ 2^(2*n+1/2) * phi^(sqrt(5)*n) * n^(5*n+5) / exp((5-sqrt(phi)*Pi/5^(1/4))*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A367823 a(n) = Product_{k=0..n} (n^6 + k^6).

Original entry on oeis.org

0, 2, 532480, 615291760980, 2759348178649088000, 38607000141072009765625000, 1421393314362014000255258433945600, 120987688662917113214734283836109985530000, 21538956798613753302336535534471908360548515840000, 7407676630843157974199152503244776880017496531101658740000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 02 2023

Keywords

Crossrefs

Cf. A126804, A272244, A272246, A272247, A272248, A324439, A367833.

Programs

  • Mathematica
    Table[Product[n^6 + k^6, {k, 0, n}], {n, 0, 10}]
  • PARI
    a(n) = prod(k=0, n, n^6 + k^6); \\ Michel Marcus, Dec 02 2023

Formula

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

A367833 a(n) = Product_{k=0..n} (n^8 + k^8).

Original entry on oeis.org

0, 2, 33685504, 3851231199376068, 2670340833224951538384896, 8442437541556000328575439453125000, 96982590530284083757788173242580213780447232, 3406021755102121469537208768079471859185253483110025744
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 02 2023

Keywords

Crossrefs

Cf. A126804, A272244, A272246, A272247, A272248, A367823.
Cf. A367834.

Programs

  • Mathematica
    Table[Product[n^8 + k^8, {k, 0, n}], {n, 0, 10}]

Formula

a(n) ~ 2^(n + 1/2) * ((2 + sqrt(2 + sqrt(2)))/(2 - sqrt(2 + sqrt(2))))^(sqrt(2 + sqrt(2))*n/2) * (sqrt(2) - 1 + sqrt(4 - 2*sqrt(2)))^(sqrt(2 - sqrt(2))*n) * exp((Pi/sqrt(2 - sqrt(2)) - 8)*n) * n^(8*(n + 1)).

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

Views

Author

Vaclav Kotesovec, Jan 01 2024

Keywords

Comments

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.

Crossrefs

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

Programs

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

Formula

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...

A367927 a(n) = Product_{k=1..n} (n^2 + k^2)^k.

Original entry on oeis.org

1, 2, 320, 9856080, 111411200000000, 758912215744345000000000, 4641055652546245989752776320614400000, 35539706227151665491451514343890494014992000000000000, 453355203871266077337924186645041169745278527392553369600000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2023

Keywords

Crossrefs

Cf. A272244.

Programs

  • Mathematica
    Table[Product[(k^2 + n^2)^k, {k, 1, n}], {n, 0, 10}]

Formula

a(n) ~ 2^(n^2 + n/2 + 1/12) * n^(n*(n+1)) / exp(n^2/2 - 1/12).
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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