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A001802 Coefficients of Legendre polynomials.

%I A001802 M3980 N1649 #31 Apr 26 2025 05:56:21
%S A001802 5,35,1260,4620,30030,90090,1021020,2771340,14549535,37182145,
%T A001802 1487285800,3650610600,17644617900,42075627300,396713057400,
%U A001802 925663800600,4281195077775,9821565178425,178970743251300,405039050516100,1822675727322450,4079321865912150
%N A001802 Coefficients of Legendre polynomials.
%D A001802 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
%D A001802 G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
%D A001802 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001802 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001802 G. C. Greubel, <a href="/A001802/b001802.txt">Table of n, a(n) for n = 0..1000</a>
%H A001802 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A001802 a(n) = 5*2^(n-4)*binomial(n+5/2, n)*2^A011371(n+6). - _G. C. Greubel_, Apr 26 2025
%t A001802 A001802[n_]:= 5*4^(n+1)*Binomial[n+5/2,n]/2^DigitCount[n+6,2,1];
%t A001802 Table[A001802[n], {n,0,30}] (* _G. C. Greubel_, Apr 26 2025 *)
%o A001802 (PARI) a(n)= - polcoeff(pollegendre(n+6), n)*2^valuation((n\2*2+6)!, 2) \\ _Michel Marcus_, May 29 2013
%o A001802 (Magma)
%o A001802 A001802:= func< n | Binomial(n+4,4)*Catalan(n+3)*2^(Valuation(Factorial(n+6),2)-n-4) >;
%o A001802 [A001802(n): n in [0..30]]; // _G. C. Greubel_, Apr 26 2025
%o A001802 (SageMath)
%o A001802 def A001802(n): return 5*2^(n-4)*binomial(n+5/2,n)*2^valuation(factorial(n+6), 2)
%o A001802 print([A001802(n) for n in range(31)]) # _G. C. Greubel_, Apr 26 2025
%Y A001802 Diagonal 4 of triangle A100258.
%Y A001802 Cf. A001796, A011371.
%K A001802 nonn
%O A001802 0,1
%A A001802 _N. J. A. Sloane_
%E A001802 More terms from _Michel Marcus_, Feb 02 2015