A349113 - OEIS

%I #11 Feb 16 2025 08:34:02

%S 1,8,40636,748832256,37759888297756,4086692369433395200,

%T 815254385427670754825764,270587150855247020644760551424,

%U 138859707622050969870951620062449436,104286590422721059977069662227099300134912,109828573459404650800550127862919905133973562480

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

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_polynomials">Legendre polynomials</a>.

%F a(n) ~ 2^(6*n) * n^(3*n - 1/2) / sqrt(3*Pi).

%t Table[8^n*LegendreP[3*n, n], {n, 0, 12}]

%o (PARI) a(n) = 8^n*pollegendre(3*n, n); \\ _Michel Marcus_, Nov 08 2021

%Y Cf. A008316, A110129, A349077, A349115.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 08 2021