Search: id:A001790
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%I A001790 M2508 N0992
%S A001790 1,1,3,5,35,63,231,429,6435,12155,46189,88179,676039,1300075,5014575,
%T A001790 9694845,300540195,583401555,2268783825,4418157975,34461632205,
%U A001790 67282234305,263012370465,514589420475,8061900920775,15801325804719
%N A001790 Numerators in expansion of 1/sqrt(1-x).
%C A001790 Also numerator of binomial(2n,n)/4^n (cf. A046161).
%C A001790 Also numerator of e(n-1,n-1) (see Maple line).
%C A001790 Leading coefficient of normalized Legendre polynomial.
%C A001790 Common denominator of expansions of powers of x in terms of Legendre 
               polynomials P_n(x).
%C A001790 Also the numerator of binomial(2n,n)/2^n. - T. D. Noe (noe(AT)sspectra.com), 
               Nov 29 2005
%C A001790 This sequence gives the numerators of the Maclaurin series of the Lorentz 
               factor (see Wikipedia link) of 1/sqrt(1-b^2)=dt/dtau where b=u/c 
               is the velocity in terms of the speed of light c, u is the velocity 
               as observed in the reference frame where time t is measured and tau 
               is the proper time. - Stephen Crowley (crow(AT)crowlogic.net), Apr 
               03 2007
%C A001790 Truncations of rational expressions like those given by the numerator 
               operator are artifacts in integer formulas and have many disadvantages. 
               A pure integer formula follows. Let n$ denotes the swinging factorial 
               and sigma(n) = number of '1's in the base 2 representation of [n/
               2]. T hen a(n) = (2*n)$ / sigma(2*n) = A056040(2*n) / A151565(2*n+1). 
               Simply said: A001790 is the odd part of the swinging factorial at 
               even indices. [From Peter Luschny (peter(AT)luschny.de), Aug 01 2009]
%D A001790 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001790 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001790 W. G. Bickley and J. C. P. Miller, Numerical differentiation near the 
               limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
%D A001790 P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, 
               p. 372.
%D A001790 V. H. Moll. The evaluation of integrals: a personal story, Notices Amer. 
               Math. Soc., 49 (No. 3, March 2002), 311-317.
%D A001790 Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, 
               Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%D A001790 H. E. Salzer, Coefficients for expressing the first twenty-four powers 
               in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.
%D A001790 Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,
               2,...,n}", preprint, April 2008. [From Peter Luschny (peter(AT)luschny.de), 
               Aug 01 2009]
%H A001790 T. D. Noe, <a href="b001790.txt">Table of n, a(n) for n=0..200</a>
%H A001790 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BinomialSeries.html">Binomial Series</a>
%H A001790 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LegendrePolynomial.html">Legendre Polynomial</a>
%H A001790 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lorentz_factor">Lorentz 
               Factor</a>.
%F A001790 a(n) = A000984(n)/A001316(n) where A001316(n) is the highest power of 
               2 dividing C(2n, n)=A000984(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jan 27 2002
%F A001790 a(n)=numerator(L(n)), with rational L(n):=binomial(2*n,n)/2^n. L(n) is 
               the leading coefficient of the Legendre polynomial P_n(x).
%F A001790 L(n)=(2*n-1)!!/n! with the double factorials (2*n-1)!!=A001147(n), n>
               =0.
%e A001790 1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/
               256, ...
%e A001790 binomial(2n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/
               2048, 6435/32768, ...
%p A001790 e := proc(l,m) local k; add(2^(k-2*m)*binomial(2*m-2*k,m-k)*binomial(m+k,
               m)*binomial(k,l),k=l..m); end;
%p A001790 Contribution from Peter Luschny (peter(AT)luschny.de), Aug 01 2009: (Start)
%p A001790 swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 
               then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p A001790 sigma := n -> 2^(add(i,i=convert(iquo(n,2),base,2))):
%p A001790 a := n -> swing(2*n)/sigma(2*n); (End)
%t A001790 Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]]
%o A001790 (PARI) a(n)=if(n<0,0,polcoeff(pollegendre(n),n)*2^valuation((n\2*2)!,
               2))
%Y A001790 Cf. A001800, A001801, A008316.
%Y A001790 First column of triangle A100258.
%Y A001790 Diagonal 1 of triangle A100258.
%Y A001790 Bisection of A036069.
%Y A001790 Cf. A005187, A060818(n)= denominator(L(n)). Bisections give A061548 and 
               A063079.
%Y A001790 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 
               2009: (Start)
%Y A001790 Cf. A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/
               2)].
%Y A001790 A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/
               2) for all values of n.
%Y A001790 (End)
%Y A001790 A163590 is the odd part of the swinging factorial, A001803 at odd indices. 
               [From Peter Luschny (peter(AT)luschny.de), Aug 01 2009]
%Y A001790 Sequence in context: A162444 A052468 A055786 this_sequence A057908 A120828 
               A077784
%Y A001790 Adjacent sequences: A001787 A001788 A001789 this_sequence A001791 A001792 
               A001793
%K A001790 nonn,easy,nice,frac
%O A001790 0,3
%A A001790 N. J. A. Sloane (njas(AT)research.att.com).

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