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.

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A203587 Vandermonde sequence using x^2 - xy + y^2 applied to (1,3,5,...,2n-1).

Original entry on oeis.org

1, 7, 2793, 173302857, 3257420083394841, 30572436265385693946286383, 213701173947351299768327343802342830897, 1552400172652063971265258662606880393824188151866063025
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Programs

  • Mathematica
    f[j_] := 2 j - 1; z = 12;
v[n_] := Product[Product[f[j]^2 - f[j] f[k] + f[k]^2,
{j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203587 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203588 *)

A203589 Vandermonde sequence using x^2 + y^2 applied to (1,3,5,...,2n-1).

Original entry on oeis.org

1, 10, 8840, 1897064000, 192924579369600000, 15340654595434137315840000000, 1423341281300698059502838358528000000000000, 215088732628531399592688671811428988579913728000000000000000
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Crossrefs

Cf. A293290, A324403.

Programs

  • Mathematica
    f[j_] := 2 j - 1; z = 12;
v[n_] := Product[Product[f[j]^2 + f[k]^2, {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203589 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203590 *)
Table[Product[(2*k - 1)^2 + (2*j - 1)^2, {k, 1, n}, {j, 1, k-1}], {n, 1, 10}] (* Vaclav Kotesovec, Sep 08 2023 *)

Formula

a(n) ~ 2^(3*n^2/2 - 3*n/2 - 3/8) * n^(n*(n-1)) / exp((6 - Pi)*n^2/4 - n + Pi/48). - Vaclav Kotesovec, Sep 08 2023

A203679 Vandermonde sequence using x^2 + xy + y^2 applied to (1,2,4,...,2^(n-1)).

Original entry on oeis.org

1, 7, 4116, 2826802944, 33920193794185101312, 110137716338572837381278474323361792, 1523742153535218423780156082312357202968690791891031031808
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Programs

  • Mathematica
    f[j_] := 2^(j - 1); 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}]          (* A203679 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203680 *)

A203681 Vandermonde sequence using x^2 - xy + y^2 applied to (1,2,4,...,2^(n-1)).

Original entry on oeis.org

1, 3, 468, 66583296, 146110606331609088, 81505592725670294610465062387712, 187767859619216944607590791201095479470807659625578496, 28804175949984280950704030973074409134657825104658343024583926573114205963127095296
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Programs

  • Mathematica
    f[j_] := 2^(j - 1); 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}]          (* A203681 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203682 *)

A203683 Vandermonde sequence using x^2 + y^2 applied to (1,2,4,...,2^(n-1)).

Original entry on oeis.org

1, 5, 1700, 601120000, 3496121614336000000, 5335507266769461885009920000000000, 34161019296423817239835748940949012820787200000000000000
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Links

Programs

  • Mathematica
    f[j_] := 2^(j - 1); z = 12;
u[n_] := Product[f[j]^2 + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203683 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203684 *)
a[n_Integer/;n>=1]:=Product[(5 4^(k (k+1)))/(4^(k+1)+1) QPochhammer[-4^-(k+1),4,k],{k,n-1}] (* Todd Silvestri, Dec 15 2014 *)

Formula

a(n) = product(((5*4^(k*(k+1)))/(4^(k+1)+1))*(-4^-(k+1);4)k, k = 1..n-1), where the q-Pochhammer symbol (c;q)_m = product(1-c*q^j, j = 0..m-1). - _Todd Silvestri, Dec 15 2014

A203687 Vandermonde sequence using x^2 - x*y + y^2 applied to (1,2,6,...,n!).

Original entry on oeis.org

1, 3, 2604, 358528427712, 12031987952968690199349362688, 370754012130869137189065686833099372581714820267297996800, 84815080863788633556908198622861272366206888854108620395778065239463085651541566677024367443968000000
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Crossrefs

Cf. A203685, A203688, A203689.

Programs

  • Mathematica
    f[j_] := j!; z = 8;
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}]          (* A203687 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203688 *)

Extensions

Definition corrected by Georg Fischer, Nov 25 2021

A203691 Vandermonde sequence using x^2 + xy + y^2 applied to the first n triangular numbers: 1,3,6,10,...,n(n+1)/2.

Original entry on oeis.org

1, 13, 35217, 106499166228, 1193900625351298928700, 125977443935710401573828500229372900, 267622663746762354024098339277838006317831656007585600
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Programs

  • Mathematica
    f[j_] := j (j + 1)/2; z = 11;
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}]          (* A203691 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203692 *)

A203693 Vandermonde sequence using x^2 - xy + y^2 applied to the first n triangular numbers: 1,3,6,10,...,n(n+1)/2.

Original entry on oeis.org

1, 7, 5859, 3201146676, 3820181459144204700, 25381616622722369500521817413900, 2012800150183968180465350145758736863679028900800, 3623820214658807704726540455788270697258988626189027049678195383091200
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Programs

  • Mathematica
    f[j_] := j (j + 1)/2; z = 11;
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}]          (* A203693 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203694 *)

A203695 Vandermonde sequence using x^2 + y^2 applied to the first n triangular numbers: 1,3,6,10,...,n(n+1)/2.

Original entry on oeis.org

1, 10, 16650, 24928779600, 111827645954579880000, 3822731887490658669088662984000000, 2132112202139633769017736141415928803125840000000000
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Programs

  • Mathematica
    f[j_] := j (j + 1)/2; z = 11;
u[n_] := Product[f[j]^2 + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203695 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203696 *)

A203697 Vandermonde sequence using x^2 + xy + y^2 applied to the Fibonacci numbers (1,2,3,5,...F(n+1)).

Original entry on oeis.org

1, 7, 1729, 102427689, 7859247487506324, 5420939118751361437801775388, 212771631278201852610030565508935087592924796, 3245346956445033097411208139940145810977419605290236367412235927744
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Programs

  • Mathematica
    f[j_] := Fibonacci[j + 1]; z = 11;
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}]          (* A203697 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203698 *)
Previous Showing 51-60 of 115 results. Next

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