%I M4407 N1861 #43 Oct 23 2024 00:40:02

%S 1,1,7,33,715,4199,52003,334305,17678835,119409675,1641030105,

%T 11435320455,322476036831,2295919134019,32968493968795,

%U 238436656380769,27767032438524099,203236010537432691,2989949596465113373

%N Coefficients of Legendre polynomials.

%C Numerators in expansion of sqrt(c(x)), c(x) the g.f. of A000108. - _Paul Barry_, Jul 12 2005

%C Coefficient of Legendre_0(x) when x^n is written in term of Legendre polynomials. - _Michel Marcus_, May 28 2013

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001795/b001795.txt">Table of n, a(n) for n = 0..100</a>

%H H. E. Salzer, <a href="http://dx.doi.org/10.1090/S0025-5718-1948-0023123-5">Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials</a>, Math. Comp., 3 (1948), 16-18.

%F 1/(sqrt(1-x) + sqrt(1+x)) = Sum_{n>=0} (a(n)/b(n))*x^(2n) where b(n) is a power of 2. - _Benoit Cloitre_, Mar 12 2002

%F For n >= 1, 2^(n+1)*a(2^(n-1)) = A001791(2^n). - _Vladimir Shevelev_, Sep 05 2010

%F a(n) = numerator(binomial(2*n-1/2, n)/(2*n+1)). - _Tani Akinari_, Oct 22 2024

%o (PARI) my(x='x+O('x^30)); apply(numerator, Vec(((1-sqrt(1-4*x))/(2*x))^(1/2))) \\ _Michel Marcus_, Feb 04 2022

%o (PARI) a(n)=numerator(binomial(2*n-1/2, n)/(2*n+1)) \\ _Tani Akinari_, Oct 22 2024

%Y Divisor of A048990 and A065097. Apparently a bisection of A002596.

%Y Bisection of A099024.

%Y Cf. A000108, A001791.

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Benoit Cloitre_, Mar 12 2002