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A001799 Coefficients of Legendre polynomials.

%I A001799 M4557 N1939 #29 Apr 25 2025 05:28:15
%S A001799 8,72,2160,15504,220248,1564920,89324640,640807200,9246847896,
%T A001799 67087213336,1957095947664,14342471475696,211153052281080,
%U A001799 1560676296310488,185256494416099008,1379131680653181504,20598677144877854232
%N A001799 Coefficients of Legendre polynomials.
%C A001799 Coefficient of Legendre_4(x) when x^n is written in term of Legendre polynomials. - _Sean A. Irvine_, Nov 28 2012
%D A001799 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001799 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001799 G. C. Greubel, <a href="/A001799/b001799.txt">Table of n, a(n) for n = 2..830</a>
%H A001799 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 A001799 a(n) = (9*(2*n)*(2*n-2)/((2*n+1)*(2*n+3)*(2*n+5))) * numerator(binomial(4*n, 2*n)/2^(4*n)). - _Sean A. Irvine_, Nov 28 2012
%p A001799 a:=n->(9*(2*n)*(2*n-2)/((2*n+1)*(2*n+3)*(2*n+5)))*numer(binomial(4*n,2*n)/2^(4*n)); # _Sean A. Irvine_, Nov 28 2012
%t A001799 A001799[n_]:= With[{B=Binomial}, 144*B[n+2,4]*Numerator[B[4*n,2*n]/2^(4*n)]/(5*B[2*n+ 5, 5])];
%t A001799 Table[A001799[n], {n,2,35}] (* _G. C. Greubel_, Apr 24 2025 *)
%o A001799 (Magma)
%o A001799 B:=Binomial;
%o A001799 A001799:= func< n | 144*B(n+2, 4)*Numerator(B(4*n, 2*n)/2^(4*n))/(5*B(2*n+5, 5)) >;
%o A001799 [A001799(n): n in [2..30]]; // _G. C. Greubel_, Apr 24 2025
%o A001799 (SageMath)
%o A001799 b=binomial
%o A001799 def A001799(n): return 144*b(n+2, 4)*numerator(b(4*n, 2*n)/2^(4*n))//(5*b(2*n+5, 5))
%o A001799 print([A001799(n) for n in range(2, 31)]) # _G. C. Greubel_, Apr 24 2025
%Y A001799 Cf. A001796.
%K A001799 nonn
%O A001799 2,1
%A A001799 _N. J. A. Sloane_
%E A001799 More terms from _Sean A. Irvine_, Nov 28 2012