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Coefficients of Legendre polynomials.
(Formerly M3116 N1263)
0

%I M3116 N1263 #35 Oct 31 2024 15:10:25

%S 1,3,27,143,3315,20349,260015,1710855,92116035,631165425,8775943605,

%T 61750730457,1755702867191,12587970424449,181858466731095,

%U 1322239639929719,154702037871777123,1137023085979691001,16789716964765636633

%N Coefficients of Legendre polynomials.

%C Numerators in expansion of c(x)^(3/2), c(x) the g.f. of A000108. - _Gerald McGarvey_, Oct 07 2008

%C Coefficient of Legendre_1(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 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 Numerators of g.f. ((1-sqrt(1-4*x))/(2*x))^(3/2). - _Sean A. Irvine_, Nov 27 2012

%F a(n) = numerator(3*binomial(2*n+1/2, n)/(2*n+3)). - _Tani Akinari_, Oct 31 2024

%o (PARI) my(x='x+O('x^30)); apply(numerator, Vec(((1-sqrt(1-4*x))/(2*x))^(3/2))) \\ _Michel Marcus_, Feb 04 2022

%o (PARI) a(n)=numerator(3*binomial(2*n+1/2, n)/(2*n+3)) \\ _Tani Akinari_, Oct 31 2024

%Y Cf. A000108.

%K nonn,changed

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, Nov 27 2012