A349114 -id:A349114 - cp's OEIS frontend

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

Paul Barry, Jul 13 2005

Central coefficients of triangle A110124.

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

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

a:= n-> LegendreP(n$2)*2^n:
seq(a(n), n=0..17);  # Alois P. Heinz, Nov 17 2024

Table[2^n LegendreP[n,n],{n,0,20}] (* Harvey P. Dale, Nov 28 2012 *)

a(n)=pollegendre(n,n)<Charles R Greathouse IV, Mar 19 2017

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

A349115 a(n) = 8^n * P(n, 3*n), where P(n, x) is n-th Legendre polynomial.

Original entry on oeis.org

1, 24, 3424, 926208, 369378816, 194988441600, 128184980586496, 100904418485993472, 92542260511611682816, 96909547417109671182336, 114095278582299648325582848, 149184455262733048487847395328, 214496285274348399077675463868416, 336346643957900669242934177071890432

Offset: 0

Vaclav Kotesovec, Nov 08 2021

nonn

In general, for k>=1, P(n, k*n) ~ 2^n * k^n * n^(n - 1/2) / sqrt(Pi).

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

Cf. A008316, A110129, A349113, A349114.

Table[8^n*LegendreP[n, 3*n], {n, 0, 15}]

a(n) = 8^n*pollegendre(n, 3*n); \\ Michel Marcus, Nov 08 2021

a(n) ~ 2^(4*n) * 3^n * n^(n - 1/2) / sqrt(Pi).

cp's OEIS Frontend

A110129 Central coefficients of a scaled Legendre triangle.

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