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%I M3123 N1266 #37 Feb 24 2023 02:33:38
%S 1,3,30,70,315,693,12012,25740,109395,230945,1939938,4056234,16900975,
%T 35102025,1163381400,2404321560,9917826435,20419054425,167890003050,
%U 344616322050,1412926920405,2893136075115,47342226683700,96742811049300,395033145117975
%N Coefficients of Legendre polynomials.
%D 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 G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A001800/b001800.txt">Table of n, a(n) for n = 0..500</a>
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>, eq. 28.
%F a(n) = (n+1) * C(2n+2, n+1) / 2^A000120(n+2).
%p wt:= proc(n) local m, r; m:=n; r:=0;
%p while m>0 do r:= r+irem(m, 2, 'm') od; r
%p end:
%p a:= n-> (n+1) *binomial(2*n+2, n+1)/2^wt(n+2):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 29 2013
%t 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 (PARI) a(n)=if(n<0,0,-polcoeff(pollegendre(n+2),n)*2^valuation((n\2*2)!,2))
%Y Cf. A001790, A001801, A008316.
%Y Cf. A001803.
%Y Diagonal 2 of triangle A100258.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_, Oct 25 2002
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