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A001790 Numerators in expansion of 1/sqrt(1-x).
(Formerly M2508 N0992)
60
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; text; 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, 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]. Then 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. - Peter Luschny, Aug 01 2009

It appears that A001790 = A060818*A001147/A000142. - James R. Buddenhagen, 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). - John M. Campbell, Jul 04 2011

a(n) = denominator of 2^n n! n!/(2*n)!. - Artur Jasinski, Nov 26 2011

a(n) = numerator of the integral 1/\pi int(1/(x^2-2x+2)^n, x=-infinity..+infinity). - Leonid Bedratyuk, Nov 17 2012

a(n) = numerator of the mean value of cos(x)^(2*n) from x=0 to 2*Pi. - Jean-François Alcover, Mar 21 2013

Also numerators in expansion of arcsin(x). - Jean-François Alcover, May 17 2013

Constant terms for normalized Legendre polynomials. - Tom Copeland, Feb 04 2016

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, Aug 01 2009]

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

Horst Alzer, Bent Fuglede, Normalized binomial mid-coefficients and power means, Journal of Number Theory, Volume 115, Issue 2, December 2005, Pages 284-294.

C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304062, 1993 (see V_n with N=1).

W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table [Annotated scanned copy]

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.

J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)

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, 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.

Numerator in (1-2t)^(-1/2)= 1 + t + (3/2)t^2 + (5/2)t^3 + (35/8)t^4 + (63/8)t^5 + (231/16)t^6 + (429/16)t^7 + ... = 1 + t + 3*t^2/2! + 15*t^3/3! + 105*t^4/4! + 945*t^5/5! + ... = e.g.f. for double factorials A001147 (cf. A094638). - Tom Copeland, Dec 04 2013

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;

# From Peter Luschny, 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)

A001790 := proc(n) binomial(2*n, n)/4^n ; numer(%) ; end proc : # R. J. Mathar, Jan 18 2013

MATHEMATICA

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

Table[Denominator[Hypergeometric2F1[1/2, n, 1 + n, -1]], {n, 0, 34}]   (* John M. Campbell, Jul 04 2011 *)

PROG

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

(Sage)

@CachedFunction

def swing(n):

    if n == 0: return 1

    return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n

A001790 = lambda n: swing(2*n)/2^A000120(2*n)

[A001790(n) for n in (0..25)]  # Peter Luschny, Nov 19 2012

CROSSREFS

Cf. A001800, A001801, A008316, A046161.

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.

From Johannes W. Meijer, 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. - Peter Luschny, Aug 01 2009

Inverse Moebius transform of A180403/A046161. - Mats Granvik, Sep 04, 2010

Sequence in context: A259853 A052468 A055786 * A173092 A057908 A120828

Adjacent sequences:  A001787 A001788 A001789 * A001791 A001792 A001793

KEYWORD

nonn,easy,nice,frac

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 27 08:32 EDT 2017. Contains 284146 sequences.