0,2

The n-th term is the denominator of the integral from 0 to Pi of (sin(x))^(2*n+1). [From James Buddenhagen (jbuddenh(AT)gmail.com), Aug 17 2008]

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).

G. Pr\'{e}vost, Tables de Fonctions Sph\'{e}riques. Gauthier-Villars, Paris, 1933, pp. 156-157.

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.

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]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Circle Line Picking

(2n+1)! / [n!^2 * 2^A000120(n)]. (n+1) * C(2n+2, n+1) / 2^[A000120(n)+1]. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 10 2004

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

a(n) = numer((2*n+1)*binomial(2*n,n)/(4^n))

(1-x)^(-3/2) = sum((2*n+1)*binomial(2*n,n)/(4^n)*x^n, n=0..infinity)

(End)

Truncations of rational expressions like those given by the numerator or denominator operators 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+1)$ / sigma(2*n+1) = A056040(2*n+1) / A151565(2*n+2). Simply said: A001803 is the odd part of the swinging factorial at odd indices. [From Peter Luschny (peter(AT)luschny.de), Aug 01 2009]

Contribution from James Buddenhagen (jbuddenh(AT)gmail.com), Aug 17 2008: (Start)

N:=50; # alter if you want more or less than 50 terms

seq(denom(int(sin(x)^(2*n+1), x=0..Pi)), n=0..N); (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+1)/sigma(2*n+1); (End)

Largest odd divisors of A001800, A002011, A002457, A005430, A033876, A086228. Bisection of A004731, A004735, A086116.

Second column of triangle A100258.

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

Cf. A001790 [(1-x)^(-1/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].

Cf. A002569 [(1-x)^(1/2)].

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, A001790 at even indices. [From Peter Luschny (peter(AT)luschny.de), Aug 01 2009]

Sequence in context: A015715 A019009 A162441 this_sequence A161738 A062741 A117561

Adjacent sequences: A001800 A001801 A001802 this_sequence A001804 A001805 A001806

nonn

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