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 1-4 of 4 results.

A203673 Vandermonde sequence using x^2 + xy + y^2 applied to (1,4,9,...,n^2).

Original entry on oeis.org

1, 21, 254163, 11213968422384, 6451450005117349260375984, 127857993263632065817610313129228311433216, 191199773886534869435599958788731398661833328276349525268803584
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Crossrefs

Cf. A367550.

Programs

  • Mathematica
    f[j_] := j^2; z = 12;
u[n_] := Product[f[j]^2 + f[j] f[k] + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203673 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203674 *)

Formula

From Vaclav Kotesovec, Nov 22 2023: (Start)
a(n) = A203012(n) * A203312(n).
a(n) ~ c * 3^(n*(3*n+1)/4) * n^(2*n^2 - 2*n - 3/2) / exp(3*n^2 - n*(n+1)*Pi*sqrt(3)/4 - 2*n), where c = Gamma(1/3)^(3/2) * 3^(7/24) * exp(Pi/(8*sqrt(3))) / (2^(5/2) * Pi^(5/2)) = 0.076580853261060033865281896312127877504662138809362419847380161198324... (End)

A367668 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^2*j^2 + j^4).

Original entry on oeis.org

1, 2704, 4343072672016, 104066856161782811235776987136, 368057974579278182597141600363036562863943425064960000, 1139317987311004502889916180807286481186277543437822119282797720728081762451885916160000
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 26 2023

Keywords

Crossrefs

Cf. A203675, A324437, A367550.

Programs

  • Mathematica
    Table[Product[Product[i^4 - i^2*j^2 + j^4, {i, 1, n}], {j, 1, n}], {n, 1, 10}]
  • Python
    from math import prod, factorial
def A367668(n): return (prod((k:=j**2)**2+(m:=i**2)*(m-k) 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 + sqrt(3))^(sqrt(3)*n*(n+1)) * n^(4*n^2 - 1) / exp(6*n^2 - Pi*n*(n+1)/2), where c = 0.219927317102868518491484945565471919409874745762951216457178735860943437...

A367670 a(n) = Product_{i=1..n, j=1..n} (i^8 + i^4*j^4 + j^8).

Original entry on oeis.org

3, 171714816, 9817265089769041882465383168, 351690857158733335833718073682368165890982417955022627663773696
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 26 2023

Keywords

Crossrefs

Cf. A367542, A367543, A367550, A367668.

Programs

  • Mathematica
    Table[Product[Product[i^8 + i^4*j^4 + j^8, {i, 1, n}], {j, 1, n}], {n, 1, 7}]
  • Python
    from math import prod, factorial
def A367670(n): return (prod((k:=j**4)**2+(m:=i**4)*(m+k) for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**4)**2*3**n # Chai Wah Wu, Nov 26 2023

Formula

a(n) = A367550(n) * A367668(n).
a(n) = A367542(n) * A367543(n) * A367668(n).
a(n) ~ c * 3^(3*n*(n+1)/2) * (2 + sqrt(3))^(sqrt(3)*n*(n+1)) * n^(8*n^2 - 2) / exp(12*n^2 - Pi*(1 + sqrt(3))*n*(n+1)/2), where c = 0.05091893538977858024246640150391280389386566805866250210433631511020673755...

A367679 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^3*j + i^2*j^2 - i*j^3 + j^4).

Original entry on oeis.org

1, 1936, 1765124816400, 19271059559619728900751360000, 25048411180596698786915756280274804766474649600000000, 23045227505577134384745253646275782332295626096040088365089618773238077194240000000000
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 26 2023

Keywords

Crossrefs

Cf. A079478, A324437, A324438, A367550, A367668.

Programs

  • Mathematica
    Table[Product[i^4 - i^3*j + i^2*j^2 - i*j^3 + j^4, {i, 1, n}, {j, 1, n}], {n, 1, 6}]
  • Python
    from math import prod, factorial
def A367679(n): return (prod(i*(i*(i*(i-j)+j**2)-j**3)+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) = A324438(n) / A079478(n).
a(n) ~ c * n^(4*n^2 - 5/6) * phi^(sqrt(5)*n*(n+1)) / exp(6*n^2 - sqrt(phi)*Pi*n*(n+1)/5^(1/4)), where phi = A001622 is the golden ratio and c = 0.2505211390193028244009922677012518708897316924498037078191143761182342931773594...
Showing 1-4 of 4 results.

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

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