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 1-3 of 3 results.

A212796 Square array read by antidiagonals: T(m,n) = number of spanning trees in C_m X C_n.

Original entry on oeis.org

1, 2, 2, 3, 32, 3, 4, 294, 294, 4, 5, 2304, 11664, 2304, 5, 6, 16810, 367500, 367500, 16810, 6, 7, 117600, 10609215, 42467328, 10609215, 117600, 7, 8, 799694, 292626432, 4381392020, 4381392020, 292626432, 799694, 8, 9, 5326848, 7839321861, 428652000000, 1562500000000, 428652000000, 7839321861, 5326848, 9
Offset: 1

Views

Author

N. J. A. Sloane, May 27 2012

Keywords

Examples

			Array begins:
  1,    2,      3,        4,          5,            6               7, ...
  2,   32,    294,     2304,      16810,       117600,         799694, ...
  3,  294,  11664,   367500,   10609215,    292626432,     7839321861, ...
  4, 2304, 367500, 42467328, 4381392020, 428652000000, 40643137651228, ...
  ...

Links

Crossrefs

Rows and columns 1..10 give A000027, A212797, A212798, A212799, A358810, A358811, A358812, A358813, A358814, A358815.
Diagonal gives A212800.
Cf. A116469, A173958, A340560.

Programs

  • Maple
    Digits:=200;
T:=(m,n)->round(Re(evalf(simplify(expand(
m*n*mul(mul( 4*sin(h*Pi/m)^2+4*sin(k*Pi/n)^2, h=1..m-1), k=1..n-1))))));
  • PARI
    default(realprecision, 120);
{T(n, k) = round(n*k*prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)))} \\ Seiichi Manyama, Jan 13 2021

Formula

T(m,n) = m*n*Prod(Prod( 4*sin(h*Pi/m)^2+4*sin(k*Pi/n)^2, h=1..m-1), k=1..n-1).

A340562 a(n) = Product_{1<=j,k<=n-1} (4*sin(j*Pi/n)^2 + 4*sin(k*Pi/n)^2).

Original entry on oeis.org

1, 8, 1296, 2654208, 62500000000, 16314248724480000, 46246966018211028668416, 1405124434459231021756179283968, 453518708737693704370173592484540645376, 1545285638496177620571506637671497728000000000000
Offset: 1

Views

Author

Seiichi Manyama, Jan 11 2021

Keywords

Crossrefs

Main diagonal of A340560.
Cf. A212800.

Programs

  • Mathematica
    Table[Product[4*Sin[j*Pi/n]^2 + 4*Sin[k*Pi/n]^2, {k, 1, n-1}, {j, 1, n-1}], {n, 1, 12}] // Round (* Vaclav Kotesovec, Feb 14 2021 *)
  • PARI
    default(realprecision, 120);
{a(n) = round(prod(j=1, n-1, prod(k=1, n-1, 4*sin(j*Pi/n)^2+4*sin(k*Pi/n)^2)))}

Formula

a(n) = A212800(n)/n^2.
a(n) ~ Gamma(1/4)^4 * exp(4*G*n^2/Pi) / (16 * Pi^3 * n^2), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

A340561 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n-1} Product_{b=1..k-1} (4*sin(a*Pi/n)^2 + 4*cos(b*Pi/k)^2) ).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 12, 16, 4, 1, 1, 29, 75, 45, 5, 1, 1, 70, 361, 384, 121, 6, 1, 1, 169, 1728, 3509, 1805, 320, 7, 1, 1, 408, 8281, 31500, 30976, 8100, 841, 8, 1, 1, 985, 39675, 284089, 508805, 261725, 35287, 2205, 9, 1
Offset: 1

Views

Author

Seiichi Manyama, Jan 11 2021

Keywords

Examples

			Square array begins:
  1, 1,   1,    1,      1,       1, ...
  1, 2,   5,   12,     29,      70, ...
  1, 3,  16,   75,    361,    1728, ...
  1, 4,  45,  384,   3509,   31500, ...
  1, 5, 121, 1805,  30976,  508805, ...
  1, 6, 320, 8100, 261725, 7741440, ...

Crossrefs

Columns 1..4 give A000012, A000027, A004146, A006235.
Rows 1..3 give A000012, A000129, A005386.
Main diagonal gives A340563.
T(n, 2*n) gives A252767.
Cf. A173958, A340476, A340560.

Programs

  • PARI
    default(realprecision, 120);
{T(n, k) = round(sqrt(prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*cos(b*Pi/k)^2))))}
Showing 1-3 of 3 results.

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

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