|
| |
|
|
A001790
|
|
Numerators in expansion of 1/sqrt(1-x).
(Formerly M2508 N0992)
|
|
38
| |
|
|
1, 1, 3, 5, 35, 63, 231, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 300540195, 583401555, 2268783825, 4418157975, 34461632205, 67282234305, 263012370465, 514589420475, 8061900920775, 15801325804719
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| 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, 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) / A060632(2*n+1). Simply said: A001790 is the odd part of the swinging factorial at even indices. [From Peter Luschny, Aug 01 2009]
It appears that A001790 = A060818*A001147/A000142 [From James Buddenhagen (jbuddenh(AT)gmail.com), Jan 20 2010]
The convolution of sequence binomial(2n,n)/4^n with itself is the constant sequence with all terms =1.
a(n) equals the denominator of Hypergeometric2F1[1/2, n, 1 + n, -1] (see Mathematica code below). [From John M. Campbell, Jul 04 2011]
a(n) = denominator of 2^n n! n!/(2*n)!. - Artur Jasinski, Nov 26 2011
|
|
|
REFERENCES
| 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.
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]
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| 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.
|
|
|
FORMULA
| 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.
|
|
|
EXAMPLE
| 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, ...
|
|
|
MAPLE
| 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)
|
|
|
MATHEMATICA
| Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]]
Table[Denominator[Hypergeometric2F1[1/2, n, 1 + n, -1]], {n, 0, 34}] (* From John M. Campbell, Jul 04 2011 *)
|
|
|
PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(pollegendre(n), n)*2^valuation((n\2*2)!, 2))
|
|
|
CROSSREFS
| 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, Aug 01 2009]
Inverse Moebius transform of A180403/A046161. [From Mats Granvik, Sep 03 2010]
Sequence in context: A162444 A052468 A055786 * A173092 A057908 A120828
Adjacent sequences: A001787 A001788 A001789 * A001791 A001792 A001793
|
|
|
KEYWORD
| nonn,easy,nice,frac
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Cross reference (Sep 03 2010) corrected by Mats Granvik (mats.granvik(AT)abo.fi), Sep 04 2010
|
| |
|
|
