0,3

Also numerator of binomial(2n,n)/4^n (cf. A046161).

Also numerator of e(n-1,n-1) (see Maple line).

Leading coefficient of normalized Legendre polynomial.

Common denominator of expansions of powers of x in terms of Legendre polynomials P_n(x).

Also the numerator of binomial(2n,n)/2^n. - T. D. Noe (noe(AT)sspectra.com), Nov 29 2005

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

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]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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).

P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372.

V. H. Moll. The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.

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]

T. D. Noe, Table of n, a(n) for n=0..200

Eric Weisstein's World of Mathematics, Binomial Series

Eric Weisstein's World of Mathematics, Legendre Polynomial

Wikipedia, Lorentz Factor.

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

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).

L(n)=(2*n-1)!!/n! with the double factorials (2*n-1)!!=A001147(n), n>=0.

1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ...

binomial(2n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ...

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;

Contribution from Peter Luschny (peter(AT)luschny.de), Aug 01 2009: (Start)

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:

sigma := n -> 2^(add(i, i=convert(iquo(n, 2), base, 2))):

a := n -> swing(2*n)/sigma(2*n); (End)

Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]]

(PARI) a(n)=if(n<0, 0, polcoeff(pollegendre(n), n)*2^valuation((n\2*2)!, 2))

Cf. A001800, A001801, A008316.

First column of triangle A100258.

Diagonal 1 of triangle A100258.

Bisection of A036069.

Cf. A005187, A060818(n)= denominator(L(n)). Bisections give A061548 and A063079.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009: (Start)

Cf. A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].

A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n.

(End)

A163590 is the odd part of the swinging factorial, A001803 at odd indices. [From Peter Luschny (peter(AT)luschny.de), Aug 01 2009]

Sequence in context: A162444 A052468 A055786 this_sequence A057908 A120828 A077784

Adjacent sequences: A001787 A001788 A001789 this_sequence A001791 A001792 A001793

nonn,easy,nice,frac

N. J. A. Sloane (njas(AT)research.att.com).