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 1-5 of 5 results.

A209330 Triangle defined by T(n,k) = binomial(n^2, n*k), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 84, 84, 1, 1, 1820, 12870, 1820, 1, 1, 53130, 3268760, 3268760, 53130, 1, 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1, 1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 06 2012

Keywords

Comments

Column 1 equals A014062.
Row sums equal A167009.
Antidiagonal sums equal A209331.
Ignoring initial row T(0,0), equals the logarithmic derivative of the g.f. of triangle A209196.

Examples

			The triangle of coefficients C(n^2,n*k), n>=k, k=0..n, begins:
1;
1, 1;
1, 6, 1;
1, 84, 84, 1;
1, 1820, 12870, 1820, 1;
1, 53130, 3268760, 3268760, 53130, 1;
1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1;
1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1; ...

Links

Crossrefs

Cf. A014062 (column 1), A167009 (row sums), A209331, A209196.
Cf. related triangles: A209196 (exp), A228836, A228832, A226234.
Cf. A206830.

Programs

  • Mathematica
    Table[Binomial[n^2, n*k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 05 2018 *)
  • PARI
    {T(n,k)=binomial(n^2,n*k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

A209428 a(n) = Sum_{k=0..[n/2]} binomial(n-k,k)^(n-k).

Original entry on oeis.org

1, 1, 2, 5, 29, 284, 4423, 146913, 12314170, 1881868883, 442540106327, 198351607585964, 242843144659704443, 641109494638274737567, 2641514784666925880476348, 17914201815999230497003603969, 302266027138470510426936352722523
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2012

Keywords

Comments

Equals antidiagonal sums of triangle A209427.

Links

Crossrefs

Cf. A209427, A209331.

Programs

  • Mathematica
    Table[Sum[Binomial[n-k,k]^(n-k),{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Mar 06 2014 *)
  • PARI
    {a(n)=sum(k=0,n\2,binomial(n-k,k)^(n-k))}
for(n=0,20,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = ((1-r)/r)^((1-r)^2/(3-4*r)) = 1.4360944969025357119535113523184471..., where r = A323777 = 0.220676041323740696312822269998... is the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r). - Vaclav Kotesovec, Mar 06 2014

A323777 Decimal expansion of the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jan 27 2019

Keywords

Examples

			0.2206760413237406963128222699980501671876810310275740395417335127215630565...

Crossrefs

Cf. A166895, A209331, A209428.
Cf. A220359, A237421, A245259, A323768, A323773, A323778.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[(1-2*r)^(3-4*r) == (1-r)^(2-2*r) * r^(1-2*r), {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A238696 a(n) = Sum_{k=0..floor(n/2)} binomial(n*(n-k), n*k).

Original entry on oeis.org

1, 1, 2, 21, 497, 18508, 3297933, 2348121769, 2319121509374, 4535739243360613, 58887253765506968848, 1694438232474931034462251, 64598311562133275526222276162, 8312693334404799592869803398802772, 5827069387752679429926992257426553147833
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 03 2014

Keywords

Links

Crossrefs

Cf. A206849, A207136, A209331.

Programs

  • Mathematica
    Table[Sum[Binomial[n*(n-k), n*k], {k, 0, Floor[n/2]}], {n, 0, 20}]
  • PARI
    a(n)=sum(k=0,n\2, binomial(n*(n-k), n*k)) \\ Charles R Greathouse IV, Jul 29 2016

Formula

Maximum is at k = n*(1-1/sqrt(5))/2 = 0.2763932... * n.
Limit n->infinity a(n)^(1/n^2) = (1+sqrt(5))/2.
Lim sup n->infinity a(n) / (5^(1/4)/(n*sqrt(2*Pi))*((1+sqrt(5))/2)^(n^2+1)) = JacobiTheta3(0,exp(-5*sqrt(5)/2)) = EllipticTheta[3,0,Exp[-5*Sqrt[5]/2]] = 1.007468786736926147579...
Lim inf n->infinity a(n) / (5^(1/4)/(n*sqrt(2*Pi))*((1+sqrt(5))/2)^(n^2+1)) = JacobiTheta2(0,exp(-5*sqrt(5)/2)) = EllipticTheta[2,0,Exp[-5*Sqrt[5]/2]] = 0.494414344263155315970...
a(n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^(2*n)) for n > 0. - Seiichi Manyama, Oct 11 2021

A228837 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2*k)*k).

Original entry on oeis.org

1, 1, 2, 5, 38, 597, 14472, 554653, 44421258, 8933194659, 3408672951784, 1984802013951149, 1803179670478111304, 3323206887194925488269, 15156709454119350064982141, 132889643918499982093215167857, 1784438297905511051093397284187186
Offset: 0

Views

Author

Paul D. Hanna, Sep 05 2013

Keywords

Comments

Equals the antidiagonal sums of triangle A228836.

Links

Crossrefs

Cf. A228836, A207138.
Cf. variants: A209331, A228833, A123165.

Programs

  • Mathematica
    Table[Sum[Binomial[(n-k)^2, (n-2*k)*k],{k,0,Floor[n/2]}],{n,0,15}] (* Vaclav Kotesovec, Sep 05 2013 *)
  • PARI
    {a(n)=sum(k=0,n\2,binomial((n-k)^2, (n-2*k)*k))}
for(n=0,30,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = ((1-r)^2/(r*(1-2*r)))^((1-3*r)*(1-r)/(3*(1-2*r))) = 1.36198508972775011599..., where r = 0.195220321930105755... is the root of the equation (1-3*r+3*r^2)^(3*(2*r-1)) = (r*(1-2*r))^(4*r-1) * (1-r)^(4*(r-1)). - Vaclav Kotesovec, added Sep 05 2013, simplified Mar 04 2014
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.