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A001798 Coefficients of Legendre polynomials.

%I A001798 M2149 N0856 #27 Apr 24 2025 08:50:24
%S A001798 2,28,182,4760,31654,428260,2941470,163761840,1152562950,16381761396,
%T A001798 117402623338,3390322778024,24634522766126,360043025043380,
%U A001798 2644479279859438,312191499849352032,2312918756095439814,34398444513178377492
%N A001798 Coefficients of Legendre polynomials.
%C A001798 Coefficient of Legendre_3(x) when x^n is written in term of Legendre polynomials. - _Sean A. Irvine_, Nov 28 2012
%D A001798 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001798 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001798 G. C. Greubel, <a href="/A001798/b001798.txt">Table of n, a(n) for n = 1..830</a>
%H A001798 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 A001798 a(n) = (14*n/((2*n+3)*(2*n+5)))*numerator(binomial(4*n+2, 2*n+1)/2^(4*n)). - _Sean A. Irvine_, Nov 28 2012
%p A001798 a:=n->(14*n/((2*n+3)*(2*n+5)))*numer(binomial(4*n+2,2*n+1)/2^(4*n)); # _Sean A. Irvine_, Nov 28 2012
%t A001798 A001798[n_]:= With[{B=Binomial}, 14*B[n+2,3]*Numerator[B[4*n+2,2*n+1]/2^(4*n) ]/B[2*n+5,4]];
%t A001798 Table[A001798[n], {n,30}] (* _G. C. Greubel_, Apr 23 2025 *)
%o A001798 (Magma)
%o A001798 B:=Binomial;
%o A001798 A001798:= func< n | 14*B(n+2,3)*Numerator(B(4*n+2,2*n+1)/2^(4*n))/B(2*n+5,4) >;
%o A001798 [A001798(n): n in [1..30]]; // _G. C. Greubel_, Apr 23 2025
%o A001798 (SageMath)
%o A001798 b=binomial
%o A001798 def A001798(n): return 14*b(n+2,3)*numerator(b(4*n+2,2*n+1)/2^(4*n) )//b(2*n+5,4)
%o A001798 print([A001798(n) for n in range(1,31)]) # _G. C. Greubel_, Apr 23 2025
%Y A001798 Cf. A001796.
%K A001798 nonn
%O A001798 1,1
%A A001798 _N. J. A. Sloane_
%E A001798 More terms from _Sean A. Irvine_, Nov 28 2012