Search: id:A001803
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%I A001803 M2986 N1207
%S A001803 1,3,15,35,315,693,3003,6435,109395,230945,969969,2028117,16900975,
%T A001803 35102025,145422675,300540195,9917826435,20419054425,83945001525,
%U A001803 172308161025,1412926920405,2893136075115,11835556670925
%N A001803 Numerators in expansion of (1-x)^(-3/2).
%C A001803 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]
%D A001803 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001803 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001803 G. Pr\'{e}vost, Tables de Fonctions Sph\'{e}riques. Gauthier-Villars, 
               Paris, 1933, pp. 156-157.
%D A001803 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.
%D A001803 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 A001803 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001803 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Heads-Minus-TailsDistribution.html">Link to a section of The World 
               of Mathematics.</a>
%H A001803 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RandomWalk1-Dimensional.html">Link to a section of The World of Mathematics.</
               a>
%H A001803 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CircleLinePicking.html">Circle Line Picking</a>
%F A001803 (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
%F A001803 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 
               2009: (Start)
%F A001803 a(n) = numer((2*n+1)*binomial(2*n,n)/(4^n))
%F A001803 (1-x)^(-3/2) = sum((2*n+1)*binomial(2*n,n)/(4^n)*x^n, n=0..infinity)
%F A001803 (End)
%F A001803 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]
%p A001803 Contribution from James Buddenhagen (jbuddenh(AT)gmail.com), Aug 17 2008: 
               (Start)
%p A001803 N:=50; # alter if you want more or less than 50 terms
%p A001803 seq(denom(int(sin(x)^(2*n+1),x=0..Pi)),n=0..N); (End)
%p A001803 Contribution from Peter Luschny (peter(AT)luschny.de), Aug 01 2009: (Start)
%p A001803 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 A001803 sigma := n -> 2^(add(i,i= convert(iquo(n,2),base,2))):
%p A001803 a := n -> swing(2*n+1)/sigma(2*n+1); (End)
%Y A001803 Largest odd divisors of A001800, A002011, A002457, A005430, A033876, 
               A086228. Bisection of A004731, A004735, A086116.
%Y A001803 Second column of triangle A100258.
%Y A001803 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 
               2009: (Start)
%Y A001803 Cf. A001790 [(1-x)^(-1/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/
               2)].
%Y A001803 Cf. A002569 [(1-x)^(1/2)].
%Y A001803 A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/
               2) for all values of n.
%Y A001803 (End)
%Y A001803 A163590 is the odd part of the swinging factorial, A001790 at even indices. 
               [From Peter Luschny (peter(AT)luschny.de), Aug 01 2009]
%Y A001803 Sequence in context: A015715 A019009 A162441 this_sequence A161738 A062741 
               A117561
%Y A001803 Adjacent sequences: A001800 A001801 A001802 this_sequence A001804 A001805 
               A001806
%K A001803 nonn
%O A001803 0,2
%A A001803 N. J. A. Sloane (njas(AT)research.att.com).

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