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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A245259 Decimal expansion of the root of the equation r^(2*r-1) = (r+1)^(2*r).

Original entry on oeis.org

3, 7, 6, 6, 7, 4, 4, 7, 4, 9, 7, 7, 2, 8, 4, 4, 9, 8, 4, 6, 9, 8, 1, 4, 8, 1, 1, 2, 8, 3, 1, 3, 0, 8, 0, 8, 5, 7, 0, 4, 1, 2, 9, 5, 2, 9, 9, 9, 4, 4, 3, 8, 6, 5, 0, 9, 9, 9, 2, 6, 9, 5, 7, 9, 5, 3, 3, 4, 0, 2, 9, 5, 6, 0, 9, 9, 5, 6, 8, 7, 1, 2, 1, 4, 6, 5, 4, 6, 1, 5, 7, 6, 9, 7, 0, 8, 4, 1, 5, 3, 0, 2, 2, 2, 7, 6
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 15 2014

Keywords

Comments

Constant is associated with A245242.

Examples

			0.3766744749772844984698148112831308085704129529994...

Crossrefs

Cf. A245242, A220359, A237421, A245260.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[r^(2*r-1) == (r+1)^(2*r), {r, 1/2}, WorkingPrecision->250], 10, 200][[1]]

A245243 Triangle, read by rows, defined by T(n,k) = C(n^2 - k^2, n*k - k^2), for k=0..n, n>=0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 28, 10, 1, 1, 455, 495, 35, 1, 1, 10626, 54264, 8008, 126, 1, 1, 324632, 10518300, 4686825, 125970, 462, 1, 1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1, 1, 553270671, 1399358844975, 11899700525790, 2254848913647, 25140840660, 30421755, 6435, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 14 2014

Keywords

Comments

Row sums equal A245242.
Central terms are A245245(n) = C(3*n^2, n^2).

Examples

			Triangle T(n,k) = C(n^2 - k^2, n*k - k^2) begins:
1;
1, 1;
1, 3, 1;
1, 28, 10, 1;
1, 455, 495, 35, 1;
1, 10626, 54264, 8008, 126, 1;
1, 324632, 10518300, 4686825, 125970, 462, 1;
1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1;
1, 553270671, 1399358844975, 11899700525790, 2254848913647, 25140840660, 30421755, 6435, 1; ...

Links

Crossrefs

Cf. A245242 (row sums), A245245 (central terms), A209330, A228832.

Programs

  • Mathematica
    Table[Binomial[n^2-k^2,n k-k^2],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jan 06 2019 *)
  • PARI
    {T(n,k) = binomial(n^2 - k^2, n*k - k^2)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
  • PARI
    {T(n,k) = binomial(n^2,n*k) * binomial(n*k,k^2) / binomial(n^2,k^2)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

T(n,k) = C(n^2, n*k) * C(n*k, k^2) / C(n^2, k^2).
T(n,k) = (n^2 - k^2)! / ( (n^2 - n*k)! * (n*k - k^2)! ).
T(n,k) = ((n+k)*(n-k))! / ( (n*(n-k))! * (k*(n-k))! ).
Showing 1-2 of 2 results.

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

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