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A001796 Coefficients of Legendre polynomials.

%I A001796 M3116 N1263 #42 May 01 2025 23:49:35
%S A001796 1,3,27,143,3315,20349,260015,1710855,92116035,631165425,8775943605,
%T A001796 61750730457,1755702867191,12587970424449,181858466731095,
%U A001796 1322239639929719,154702037871777123,1137023085979691001,16789716964765636633
%N A001796 Coefficients of Legendre polynomials.
%C A001796 Numerators in expansion of c(x)^(3/2), c(x) the g.f. of A000108. - _Gerald McGarvey_, Oct 07 2008
%C A001796 Coefficient of Legendre_1(x) when x^n is written in term of Legendre polynomials. - _Michel Marcus_, May 28 2013
%D A001796 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001796 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001796 G. C. Greubel, <a href="/A001796/b001796.txt">Table of n, a(n) for n = 0..830</a>
%H A001796 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 A001796 Numerators of g.f. ((1-sqrt(1-4*x))/(2*x))^(3/2). - _Sean A. Irvine_, Nov 27 2012
%F A001796 a(n) = numerator(3*binomial(2*n+1/2, n)/(2*n+3)). - _Tani Akinari_, Oct 31 2024
%t A001796 Table[Numerator[3*Binomial[2*n+1/2, n]/(2*n+3)], {n,0,30}] (* _G. C. Greubel_, Apr 23 2025 *)
%o A001796 (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 A001796 (PARI) a(n)=numerator(3*binomial(2*n+1/2, n)/(2*n+3)) \\ _Tani Akinari_, Oct 31 2024
%o A001796 (Magma)
%o A001796 A001796:= func< n | Numerator(3*(n+1)*Catalan(2*n+1)/(4^n*(2*n+3))) >;
%o A001796 [A001796(n): n in [0..25]]; // _G. C. Greubel_, Apr 23 2025
%o A001796 (SageMath)
%o A001796 def A001796(n): return numerator(3*binomial(2*n+1/2, n)/(2*n+3))
%o A001796 print([A001796(n) for n in range(31)]) # _G. C. Greubel_, Apr 23 2025
%Y A001796 Cf. A001795, A001797, A001798, A001799, A001800, A001801, A001802.
%Y A001796 Cf. A000108.
%K A001796 nonn
%O A001796 0,2
%A A001796 _N. J. A. Sloane_
%E A001796 More terms from _Sean A. Irvine_, Nov 27 2012