A001803 Numerators in expansion of (1 - x)^(-3/2).
1, 3, 15, 35, 315, 693, 3003, 6435, 109395, 230945, 969969, 2028117, 16900975, 35102025, 145422675, 300540195, 9917826435, 20419054425, 83945001525, 172308161025, 1412926920405, 2893136075115, 11835556670925
Offset: 0
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.
- G. Prévost, Tables de Fonctions Sphériques. 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).
- Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:9 at page 51.
Links
- T. D. Noe, Table of n, a(n) for n = 0..200
- Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- Mats Erik Andersson, Das Flaviussche Sieb, Acta Arith., 85 (1998), 301-307.
- Alexander Barg, Stolarsky's invariance principle for finite metric spaces, arXiv:2005.12995 [math.CO], 2020.
- Yue-Wu Li and Feng Qi, A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
- Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
- Eric Weisstein's World of Mathematics, Heads-Minus-Tails Distribution, Random Walk--1-Dimensional, Circle Line Picking.
Crossrefs
The denominator is given in A046161.
Second column of triangle A100258.
Cf. A002596 (numerators in expansion of (1-x)^(1/2)).
Cf. A161198 (triangle related to the series expansions of (1-x)^((-1-2*n)/2)).
A163590 is the odd part of the swinging factorial, A001790 at even indices. - Peter Luschny, Aug 01 2009
Programs
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Julia
A001803(n) = sum(<<(A001790(k), A005187(n) - A005187(k)) for k in 0:n) # Peter Luschny, Oct 03 2019
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Magma
A001803:= func< n | Numerator(Binomial(n+2,2)*Catalan(n+1)/4^n) >; [A001803(n): n in [0..30]]; // G. C. Greubel, Apr 27 2025
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Maple
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); # Peter Luschny, Aug 01 2009 A001803 := proc(n) (2*n+1)*binomial(2*n,n)/4^n ; numer(%) ; end proc: # R. J. Mathar, Jul 06 2011 a := n -> denom(Pi*binomial(n, -1/2)): seq(a(n), n = 0..22); # Peter Luschny, Dec 06 2024
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Mathematica
Numerator/@CoefficientList[Series[(1-x)^(-3/2),{x,0,25}],x] (* Harvey P. Dale, Feb 19 2011 *) Table[Denominator[Beta[1, n + 1, 1/2]], {n, 0, 22}] (* Gerry Martens, Nov 13 2016 *) -
PARI
a(n) = numerator((2*n+1)*binomial(2*n,n)/(4^n)); \\ Altug Alkan, Sep 06 2018
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SageMath
def A001803(n): return numerator((n+1)*binomial(2*n+2,n+1)/2^(2*n+1)) print([A001803(n) for n in range(31)]) # G. C. Greubel, Apr 27 2025
Formula
a(n) = (2*n + 1)! /(n!^2*2^A000120(n)) = (n + 1)*binomial(2*n+2,n+1)/2^(A000120(n)+1). - Ralf Stephan, Mar 10 2004
From Johannes W. Meijer, Jun 08 2009: (Start)
a(n) = numerator( (2*n+1)*binomial(2*n,n)/(4^n) ).
(1 - x)^(-3/2) = Sum_{n>=0} ((2*n+1)*binomial(2*n,n)/4^n)*x^n. (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$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n+1)$ / sigma(2*n+1) = A056040(2*n+1) / A060632(2*n+2). Simply said: This sequence gives the odd part of the swinging factorial at odd indices. - Peter Luschny, Aug 01 2009
a(n) = denominator(Pi*binomial(n, -1/2)). - Peter Luschny, Dec 06 2024
Comments