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 1-2 of 2 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

A115951 Expansion of 1/sqrt(1-4*x*y-4*x^2*y).

Original entry on oeis.org

1, 0, 2, 0, 2, 6, 0, 0, 12, 20, 0, 0, 6, 60, 70, 0, 0, 0, 60, 280, 252, 0, 0, 0, 20, 420, 1260, 924, 0, 0, 0, 0, 280, 2520, 5544, 3432, 0, 0, 0, 0, 70, 2520, 13860, 24024, 12870, 0, 0, 0, 0, 0, 1260, 18480, 72072, 102960, 48620, 0, 0, 0, 0, 0, 252, 13860, 120120, 360360, 437580, 184756
Offset: 0

Views

Author

Paul Barry, Mar 14 2006

Keywords

Comments

Row sums are A006139. Diagonal sums are A115962.
Coefficients of 2^n * P(n, x) with P the Legendre P polynomials. Reflection of triangle A008556. - Ralf Stephan, Apr 07 2016.

Examples

			Triangle begins
   1,
   0,  2,
   0,  2,  6,
   0,  0, 12,  20,
   0,  0,  6,  60,  70,
   0,  0,  0,  60, 280,  252,
   0,  0,  0,  20, 420, 1260, 924

Links

Crossrefs

Cf. A006139 (row sums), A063007 (binomial transform), A115962 (diagonal sums).

Programs

  • Magma
    /* As triangle */ [[Binomial(2*k,k)*Binomial(k,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 03 2015
  • Mathematica
    Table[Binomial[2k, k]Binomial[k, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* Michael De Vlieger, Sep 02 2015 *)
  • PARI
    {T(n,k) = binomial(2*k,k)*binomial(k,n-k)}; \\ G. C. Greubel, May 06 2019
  • Sage
    [[binomial(2*k,k)*binomial(k,n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 06 2019

Formula

Number triangle T(n,k) = C(2k,k)*C(k,n-k).
From Peter Bala, Sep 02 2015: (Start)
Binomial transform is A063007; equivalently, P * M = A063007, where P denotes Pascal's triangle A007318 and M denotes the present array.
P * M * P^-1 is a signed version of A063007. (End)
Showing 1-2 of 2 results.

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

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