A110129 Central coefficients of a scaled Legendre triangle.

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

Offset: 0

Author

Paul Barry, Jul 13 2005

Keywords

• easy
• nonn

Comments

Central coefficients of triangle A110124.

Links

• Eric Weisstein's World of Mathematics, Legendre Polynomial.
• Wikipedia, Legendre polynomials.

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