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A001803 Numerators in expansion of (1-x)^(-3/2).
(Formerly M2986 N1207)
37
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; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is the denominator of the integral from 0 to Pi of (sin(x))^(2*n+1). [James R. Buddenhagen, 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) = (2n+1)*A001790(n). A046161(n)/a(n) = 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+... ). See A173384 and A173396. [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, 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

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. [Peter Luschny, Aug 01 2009]

MAPLE

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]  (* 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.

From Johannes W. Meijer, Jun 08 2009: (Start)

Cf. A001790, A161199  and A161201.

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. [Peter Luschny, Aug 01 2009]

Sequence in context: A187787 A019009 A162441 * A161738 A062741 A185541

Adjacent sequences:  A001800 A001801 A001802 * A001804 A001805 A001806

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 18 07:04 EDT 2014. Contains 240706 sequences.