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.

A110129 Central coefficients of a scaled Legendre triangle.

Original entry on oeis.org

1, 2, 22, 504, 16966, 752800, 41492284, 2734083968, 209681631814, 18348172005888, 1804161160185748, 196945525458761728, 23633625832975567644, 3092337510752711057408, 438161926888980929318584, 66838962347916069718425600, 10921339483491720513675526726, 1903085098399657078831752282112
Offset: 0

Views

Author

Paul Barry, Jul 13 2005

Keywords

Comments

Central coefficients of triangle A110124.

Links

Crossrefs

Cf. A008556, A110124, A349077, A349113, A349114, A349115.

Programs

  • Maple
    a:= n-> LegendreP(n$2)*2^n:
seq(a(n), n=0..17);  # Alois P. Heinz, Nov 17 2024
  • Mathematica
    Table[2^n LegendreP[n,n],{n,0,20}] (* Harvey P. Dale, Nov 28 2012 *)
  • PARI
    a(n)=pollegendre(n,n)<Charles R Greathouse IV, Mar 19 2017

Formula

a(n) = 2^n*LegendreP(n, n).
a(n) = Sum_{j=0..floor(n/2)} (-1)^j*C(n, j)*C(2*n-2*j, n)*n^(n-2*j).
a(n) ~ 2^(2*n) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Nov 07 2021

A110125 Row sums of a scaled Legendre triangle.

Original entry on oeis.org

1, 1, 1, 9, 43, 213, 1507, 12537, 114325, 1144221, 12512373, 148351881, 1892798733, 25816700309, 374406371121, 5748436717505, 93098093268755, 1585364420710205, 28305454044441843, 528500175862180033, 10295596472841577779
Offset: 0

Views

Author

Paul Barry, Jul 13 2005

Keywords

Comments

Row sums of A110124.

Links

Programs

  • Maple
    f := n -> add(2^(n-k)*orthopoly[P](n-k,k),k=0..n):
map(f, [$0..30]); # Robert Israel, Jun 17 2019
  • PARI
    a(n)=sum(k=0,n, pollegendre(n-k, k)<<(n-k)) \\ Charles R Greathouse IV, Mar 19 2017

Formula

a(n)=sum{k=0..n, 2^(n-k)*LegendreP(n-k, k)}.

A110126 Diagonal sums of a scaled Legendre triangle.

Original entry on oeis.org

1, 0, -1, 2, 11, 12, 19, 174, 1073, 6584, 45871, 337098, 2563293, 20346204, 169904575, 1501012310, 14065496355, 139601710224, 1460544299739, 16007830782402, 182743810845223, 2163753452707580, 26506423427738891, 335585442530912862, 4389743577358599363
Offset: 0

Views

Author

Paul Barry, Jul 13 2005

Keywords

Comments

Diagonal sums of A110124.

Formula

a(n)=sum{k=0..floor(n/2), 2^(n-2k)*LegendreP(n-2k, k)}.

A110130 Central coefficients of a scaled Legendre triangle.

Original entry on oeis.org

1, 0, 2, 4, 22, 136, 504, 5136, 16966, 240368, 752800, 13660480, 41492284, 918172848, 2734083968, 71399743744, 209681631814, 6312095369056, 18348172005888, 625558677709824, 1804161160185748, 68713071248064560
Offset: 0

Views

Author

Paul Barry, Jul 13 2005

Keywords

Comments

Central coefficients of triangle A110124.

Programs

Formula

a(n)=2^ceiling(n/2)*LegendreP(ceiling(n/2), floor(n/2));
Showing 1-4 of 4 results.

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

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