|
| |
|
|
A001803
|
|
Numerators in expansion of (1-x)^(-3/2).
(Formerly M2986 N1207)
|
|
32
| |
|
|
1, 3, 15, 35, 315, 693, 3003, 6435, 109395, 230945, 969969, 2028117, 16900975, 35102025, 145422675, 300540195, 9917826435, 20419054425, 83945001525, 172308161025, 1412926920405, 2893136075115, 11835556670925
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| a(n) 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]
a(n) is the denominator of (2n)!!/(2n+1)!! = 2^(2*n)*n!*n!/(2*n+1)! (see Andersson). - N. J. A. Sloane, Jun 27 2011
a(n) = A005408*A001790. A046161/A001803 = 1, 2/3, 8/15, 16/35, 128/315, 256/693, ... is binomial transform of Madhava-Gregory-Leibniz series for Pi/4 (1-1/3+1/5-1/7+... ), from A005408 signed. See A173384 and A173396. [From Paul Curtz, Feb 21 2010]
|
|
|
REFERENCES
| 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.
M. E. Andersson, Das Flaviussche Sieb, Acta Arith., 85 (1998), 301-307.
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]
G. Pr\'{e}vost, Tables de Fonctions Sph\'{e}riques. Gauthier-Villars, Paris, 1933, pp. 156-157.
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
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, Heads-Minus-Tails Distribution, Random Walk--1-Dimensional, Circle Line Picking
|
|
|
FORMULA
| (2n+1)! / [n!^2 * 2^A000120(n)]. (n+1) * C(2n+2, n+1) / 2^[A000120(n)+1]. - Ralf Stephan, Mar 10 2004
Contribution from Johannes W. Meijer, 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) / A060632(2*n+2). Simply said: A001803 is the odd part of the swinging factorial at odd indices. [From Peter Luschny, Aug 01 2009]
|
|
|
MAPLE
| Contribution 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+1)/sigma(2*n+1); (End)
A001803 := proc(n) (2*n+1)*binomial(2*n, n)/4^n ; numer(%) ; end proc: # R. J. Mathar, Jul 06 2011
|
|
|
MATHEMATICA
| Numerator/@CoefficientList[Series[(1-x)^(-3/2), {x, 0, 25}], x] (* From Harvey P. Dale, Feb 19 2011 *)
|
|
|
CROSSREFS
| 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, Jun 08 2009: (Start)
Cf. A001790 [(1-x)^(-1/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].
Cf. A002596 [(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, Aug 01 2009]
Sequence in context: A015715 A019009 A162441 * A161738 A062741 A185541
Adjacent sequences: A001800 A001801 A001802 * A001804 A001805 A001806
|
|
|
KEYWORD
| nonn,frac
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
