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 1-7 of 7 results.

A323542 a(n) = Product_{k=0..n} (k^4 + (n-k)^4).

Original entry on oeis.org

0, 1, 512, 1896129, 14101250048, 242755875390625, 7888809923487203328, 452522453429009743939201, 42521926771106843499966758912, 6212193882217859346149080691430849, 1350441156698962215630405632000000000000, 421551664651621436548685508587919503984205889
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Links

Crossrefs

Cf. A272247, A323497, A323540, A323541, A323543, A323544, A323545, A323546.
Cf. 2*A000538 and A259108 (with sum instead of product).

Programs

  • Magma
    [(&*[(k^4 + (n-k)^4): k in [0..n]]): n in [0..15]]; // Vincenzo Librandi, Jan 18 2019
  • Mathematica
    Table[Product[k^4+(n-k)^4, {k, 0, n}], {n, 0, 15}]
  • PARI
    m=4; vector(15, n, n--; prod(k=0,n, k^m + (n-k)^m)) \\ G. C. Greubel, Jan 18 2019
  • Sage
    m=4; [product(k^m +(n-k)^m for k in (0..n)) for n in (0..15)] # G. C. Greubel, Jan 18 2019

Formula

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

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

Views

Author

Vaclav Kotesovec, Feb 28 2019

Keywords

Crossrefs

Cf. A079478, A324403, A324426, A324438, A324439, A324440, A367834.
Cf. A203677, A272247, A307215.

Programs

  • Maple
    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
  • Mathematica
    Table[Product[i^4 + j^4, {i, 1, n}, {j, 1, n}], {n, 1, 6}]
  • Python
    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

Formula

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

Extensions

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

A272244 a(n) = Product_{k=0..n} (n^2 + k^2).

Original entry on oeis.org

0, 2, 160, 21060, 4352000, 1313845000, 547573478400, 301758856490000, 212663770808320000, 186659516597629140000, 199722414913149440000000, 255947740845844788169000000, 387074162712817024892928000000, 682170272459193898736228210000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 23 2016

Keywords

Crossrefs

Cf. A126804, A272246, A272247, A272248, A367823, A367833.
Cf. A101686, A272180, A323540, A324403.

Programs

  • Mathematica
    Table[Product[n^2+k^2,{k,0,n}],{n,0,15}]

Formula

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

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).

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)).
Showing 1-7 of 7 results.

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