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A001800 Coefficients of Legendre polynomials.

%I A001800 M3123 N1266 #42 Apr 25 2025 05:22:38
%S A001800 1,3,30,70,315,693,12012,25740,109395,230945,1939938,4056234,16900975,
%T A001800 35102025,1163381400,2404321560,9917826435,20419054425,167890003050,
%U A001800 344616322050,1412926920405,2893136075115,47342226683700,96742811049300,395033145117975
%N A001800 Coefficients of Legendre polynomials.
%D A001800 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 A001800 G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
%D A001800 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001800 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001800 Alois P. Heinz, <a href="/A001800/b001800.txt">Table of n, a(n) for n = 0..500</a>
%H A001800 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].
%H A001800 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>, eq. 28.
%F A001800 a(n) = (n+1) * C(2n+2, n+1) / 2^A000120(n+2).
%p A001800 wt:= proc(n) local m, r; m:=n; r:=0;
%p A001800        while m>0 do r:= r+irem(m, 2, 'm') od; r
%p A001800      end:
%p A001800 a:= n-> (n+1) *binomial(2*n+2, n+1)/2^wt(n+2):
%p A001800 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 29 2013
%t A001800 a[n_] := (n+1)*Binomial[2*n+2, n+1]/2^DigitCount[n+2, 2, 1]; Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Mar 13 2014 *)
%o A001800 (PARI) a(n)=if(n<0,0,-polcoeff(pollegendre(n+2),n)*2^valuation((n\2*2)!,2))
%o A001800 (Magma)
%o A001800 A001800:= func< n | (n+1)*(n+2)*Catalan(n+1)/2^(&+Intseq(n+2, 2)) >;
%o A001800 [A001800(n): n in [0..30]]; // _G. C. Greubel_, Apr 25 2025
%o A001800 (SageMath)
%o A001800 def A001800(n): return (n+1)*binomial(2*n+2,n+1)//2^sum((n+2).digits(2))
%o A001800 print([A001800(n) for n in range(31)]) # _G. C. Greubel_, Apr 25 2025
%Y A001800 Cf. A001790, A001796, A001803, A008316.
%Y A001800 Diagonal 2 of triangle A100258.
%K A001800 nonn
%O A001800 0,2
%A A001800 _N. J. A. Sloane_
%E A001800 More terms from _Michael Somos_, Oct 25 2002